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ICFAI

Formulae And Tables

The Icfai University Press

ICFAI

Formulae and Tables

Board of EditorS : Prof. V R K Chary, CFA

Mr. S Sarkar, CFA

Mr. Prakash Bhattacharya, CFA

Ms. V D M V Lakshmi, CFA

ISBN : 81-7881-261-4

©The ICFAI University, All rights reserved.

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ICFAI

Contents

Preface

A. Formulae

Section I: Actuarial Principles and Practice 1

• Measurement of Interest 1

• Introduction to Annuities 2

• Demography 8

• Survival Models 9

• Mortality Tables 9

• Assurance and Annuity Benefits 10

• Premiums for Assurance and Annuity Plans 13

• Credibility Theory 16

• Loss Distributions and Risk Models 17

• Policy Values 19

• Surplus and it’s Distribution 19

Section II: Economics 20

• Supply and Demand Analysis 20

• Consumer Behavior and Analysis 21

• Production Analysis 22

• Analysis of Costs 22

• Market Structure: Perfect Competition 23

• Market Structure: Monopoly 24

• Market Structure: Oligopoly 24

• Measurement of Macro Economic Aggregates 24

• The Simple Keynesian Model of Income Determination 25

• Income Determination Model including Money and Interest 25

• Money Supply and Banking System 26

• The Open Economy and Balance of Payments 26

• Modern Macro Economics: Fiscal Policy, 27

Budget Deficits and Government Debt

Section III: Financial Management 28

• Time Value of Money 28

• Risk and Return 29

• Valuation of Securities 31

• Financial Statement Analysis 32

• Financial Forecasting 33

• Leverages 34

ICFAI

• Cost of Capital 35

• Capital Structure 36

• Dividend Policy 37

• Estimation of Working Capital Needs 38

• Inventory Management 39

• Receivables Management 39

• Cash Management 41

• Capital Expenditure Decisions 41

Section IV: Financial Risk Management 43

• Corporate Risk Management 43

• Futures 43

• Options 45

• Swaps 48

• Sensitivity of Option Premiums 48

• Value at Risk 49

Section V: International Finance 50

• The Foreign Exchange Market 50

• Exchange Rate Determination 50

• International Project Appraisal 51

• International Equity Investments 51

• Short-term Financial Management 52

Section VI: Investment Banking and Financial Services 53

• Money Market 53

• Rights Issues 53

• Lease Evaluation 53

• Hire Purchase 55

• Consumer Credit 56

• Housing Finance 57

• Venture Capital 57

Section VII: Management Accounting 58

• Cost-Volume-Profit Analysis 58

• Standard Costing and Variance Analysis 59

Section VIII: Portfolio Management 61

• Capital Market Theory 61

• Arbitrage Pricing Theory (APT Model) 61

• Asset Allocation 62

• Delineating Efficient Frontiers 62

• Portfolio Analysis 62

ICFAI

• Portfolio Performance 64

• Bond Portfolio Management 65

Section IX: Project Management 66

• Appraisal Criteria 66

• Risk Analysis in Capital Investment Decisions 66

• Application of Portfolio Theories in Investment Risk Appraisal 67

• Social Cost Benefit Analysis 67

• Options in Investment Appraisal 67

• Project Scheduling 68

• Project Monitoring and Control 68

Section X: Quantitative Methods 70

• Basics of Mathematics 70

• Calculus 70

• Interpolation and Extrapolation 72

• Central Tendency and Dispersion 73

• Probability 75

• Probability Distribution and Decision Theory 76

• Statistical Inferences 78

• Simple Linear Regression and Correlation 79

• Multiple Regression 80

• Time Series Analysis 80

• Index Numbers 81

• Quality Control 82

• Chi-Square Test and Analysis of Variance 83

Section XI: Security Analysis 85

• Bond Valuation 85

• Equity Stock Valuation Model 87

• Technical Analysis 87

• Warrants and Convertibles 88

• Real Assets and Mutual Funds 88

Section XII: Strategic Financial Management 89

• Capital Structure 89

• Decisions Support Models 89

• Working Capital Management 90

• Firms in Financial Distress 91

• Valuation of Firms 91

• Mergers and Acquisitions 91

ICFAI

B. Tables 93

1. Interest Rate Tables: 95

• Future Value Interest Factor 95-96

• Future Value Interest Factor for an Annuity 97-98

• Present Value Interest Factor 99-100

• Present Value Interest Factor for an Annuity 101-102

2. Standard Normal Probability Distribution Table 103

3. t Distribution Table 104

4. Area in the Right Tail of a Chi-Square (χ2) Distribution Table 105-106

5. F Distribution Table 107-108

6. Control Chart Factors Table 109

7. Table for Value of Call Option as Percentage of Share Price 110-111

8. Table for N(x) 112-113

9. Table for Relationship between Nominal and Effective Rates of

Interest and Discount

114-115

C. Formulae Index 116

ICFAI

FORMULAE

ICFAI

I. Actuarial Principles and Practice

1. Measurement of Interest

i. Future Value of a lump sum (Single Flow)

FVn = PV(1 + i) n

Where,

FVn = Future value of the initial flow n years hence

PV = Initial cash flow

i = Annual Rate of Interest

n = Life of investment

ii. Doubling Period = 0.35 +69

Interest Rate

iii. Future value of a lump sum with increased frequency of compounding

FVn = PV(1 +i

m) m n×

Where,

FVn = Future value after ‘n’ years

PV = Cash flow today

i = Nominal Interest Rate per Annums

m = Number of times compounding is done during a year

n = Number of years for which compounding is done

iv. The relationship between Effective vs. Nominal Rate of Interest

r =mi

(1+ )m

– 1

Where,

r = Effective rate of interest

i = Nominal rate of interest

m = Frequency of compounding per year

v. Accumulated value of an Annuity

FVAn = An(1 i) 1

i

+ −

= n

s

Where,

FVAn = Accumulation at the end of n years

A = Amount deposited/invested at the end of every year for n years

i = Rate of interest (expressed in decimals)

n = Time horizon or number of installments

sn

= Accumulated value of an annuity

vi. Sinking Fund factor = n

i

(1 i) 1

+ −

Where,

i = Rate of interest

n = Number of years

ICFAI

Formulae and Tables

2

vii. Present Value Interest Factor of an Annuity, n

a = n

n

(1 i) 1

i(1 i)

+ −

+

Where,


n = Number of years

viii. Capital Recovery Factor

A = n

n

i(1 i)

(1 i) 1

+

+ −

Where,


n = Number of years

ix. Present Value of a Perpetuity

a∞

= 1

i

Where,

i = Rate of interest.

2. Introduction to Annuities

i. Present Value of an Immediate Annuity Certain, n

a = n(1 v )

i

−

Where,

n

a = Present value of an Annuity

nv = Present value of the nth payment payable at the end of the nth year

= 1/(1 + i) n

ii. Present Value of a Deferred Annuity Certain = n

m a = mv

na

Where,

m = Deferment period

v = 1

1 i+


iii. Accumulated Value of a Deferred Annuity Certain, m

n(1 + i) s

Where,

m = Deferment period

n = Number of Annuity Installments


n

s = Accumulated value of an Annuity

iv. Present Value of an Annuity Due, n

a&& = ( )n

1 i a+

Where,

n

a = Present value of an Immediate Annuity Certain

n = The number of annuity installments

i = The rate of interest

ICFAI

Actuarial Principles and Practice

3

v. Accumulated Value of an Annuity Due, n

s&& = n

(1 + i)s

Where,

n

s = Present value of an Immediate Annuity Certain

n = The number of annuity installments


vi. Present value of a deferred annuity due of Re. one p.a. for a term of n years certain

and the deferment period is being m years

= m an

&& = m

v an

&&

Where,

v = 1

1 i+


an

&& = Present value of an Annuity due

vii. Accumulated value of a deferred annuity due of Re. one p.a. for a term of n years

certain and the deferment period is being m years

= n n

m s (1 + i) s=&&

Where,


n

s = The accumulated value of an annuity

viii. Present value of an immediate perpetuity, 1

ai∞

=

Where,


ix. Present value of a perpetuity due, 1

ad∞

=&&

Where,

d = The rate of discounting = v.i = i

1 i+

x. Present value of a deferred Perpetuity with deferment period of m years, where the

first payment is to be made immediately on completion of m years

= m a∞

&& = m 1v

i

−

Where,


v = 1

1 i+

xi. Present value of a deferred Perpetuity with deferment period of m years, where first

payment is made one year after completion of m years m

v

i

Where,


v = 1

1 i+

ICFAI

Formulae and Tables

4

xii. Present Value of an Immediate Increasing Annuity

a. n

(Ia) = n

na nv i

− && =

n

n

n

a nva

i

−+

Where,

n

a&& = The present value of an annuity due

n

a = The present value of an annuity certain

n = Number of installments


v = 1

1 i+

b. Present value of an increasing annuity due ( )

n

n

nn

a nvIa a

i

−= +

&&&& &&

Where,

n

a&& = The present value of an annuity due



v = 1

1 i+

c. Accumulated value of an increasing annuity due

( )( )

n

nn

s n 1 iIs s

i

− × += +

&&&& &&

Where,

n

s&& = The Accumulated value of an annuity due



xiii. Present Value of an Immediate Increasing Perpetuity, ( )2

1 1Ia

i i∞

= +

Where,


xiv. Present Value of an Increasing Perpetuity Due, (I a&& )∞

= 2

1

d

Where,

d = The rate of discounting = i

1 i+


ICFAI


5

xv. The Present Value of an Increasing Annuity wherein the consecutive periodical

annuity payments are in an Arithmetic Progression = An

a + D

n

na nv

i

−

Where,

A = The payment at the end of first year

D = The common difference

n

a = The present value of an Annuity certain

n = The number of installments

v = 1

1 i+


xvi. The Present Value of an Increasing Annuity wherein the consecutive periodical

annuity payments are in a Geometric Progression

= An n1 R v

(1 i) R

−

+ −

Where,

v = 1

1 i+

R = The common multiple



A = The amount of first installment

xvii. Accumulated Value of Increasing Immediate Annuity by Re. One per annum

= (Is)n

= n

s +n

s n

i

−

Where,

n

s = Accumulated value of an Annuity certain

n = Number of annuity installments


xviii. The Accumulated Value of an Increasing Annuity wherein the consecutive

periodical annuity payments are in an Arithmetic Progression

= A.n

s + D ns n

i

−

Where,


D = The amount of common difference


n

s = The accumulated value of an annuity certain


ICFAI

Formulae and Tables

6

xix. The Accumulated Value of an Increasing Immediate Annuity wherein the

consecutive periodical annuity payments are in a Geometric Progression

( )

n n1 i RA

(1 i) R

+ −

+ −

Where,


R = The common ratio



xx. Present Value of an Immediate Annuity of Re.1 p.a. for a term of n years under

which payments are made p times a year

(p)

na =

n (p)

ia

i×

Where,

i = The rate of interest per annum

n


i(p)

= p(1+ i) 1 × p −

xxi. Accumulated Value of an Immediate Annuity of Re.1 p.a. for a term of n years

under which payments are made p times a year

(p)

ns =

ns

(p)

i

i

× v

Where,


v = 1

1 i+

n

s = The Accumulated value of an Annuity certain

i(p)

= p(1+ i) 1 × p −

xxii. Present Value of an Annuity Due of Re. 1 p.a. for n years under which payments are

made ‘p’ times a year

(p)

na&& =

n (p)

i ia

pi

+

Where,


n


i(p)

= p(1+ i) 1 × p −

ICFAI


7

xxiii. Accumulated Value of an Annuity Due of Re.1 p.a. for n years under which

payments are made ‘p’ times a year

(p)

ns&& =

ns

(p)

i i

pi

+

Where,


n

s = The accumulated value of an Annuity certain

i(p)

= p(1+ i) 1 × p −

xxiv. An immediate annuity for n years where payment of ‘r’ are made at each interval of

‘r’ years, n being an exact multiple of ‘r’ and the number of payments being n

r

a. Present value of the above Annuity = (1/ r) n

nr

raa

s=

Where,

n

a = The present value of an Annuity certain for n years

r

s = The accumulated value of an Annuity for r years

b. Accumulated value of the above annuity (1/ r) n

nr

rss

s=

Where,

n

s = The present value of an Annuity certain for n years

r

s = The accumulated value of an Annuity for r years

xxv. Present value and accumulated value of an annuity due for n years where payments

of ‘r’ are made at interval of ‘r’ years, n being exact multiple of ‘r’

a. Present value: a

(1/ r) n

n ar

a r=&&

Where,

n

a = The present value of an Annuity certain for n years

r

a = The present value of an Annuity for r years

b. Accumulated value: s

(1/ r) n

n ar

s r=&&

Where,

n

s = The present value of an Annuity certain for n years

r

a = The present value of an Annuity for r years

xxvi. Capital Redemption Policies

a. The Amount of Annual Premium

n

1n 1

1P

s −+

=

Where,

n 1

s+

= The Accumulated value of an Annuity certain for a period of

n + 1 years at a rate of interest of i per annum

ICFAI

Formulae and Tables

8

b. Single Premium n

n n

1A v

(1 i)= =

+

Where,


n = The number of years

xxvii. Average Interest Yield on the Life Fund = 2I

A B I+ −

Where,

A = The fund at the beginning

B = The fund at the end of the year

I = The interest earned during the year after payment of tax

xxviii. Office premium n n nA A l A′ ′= +

Where,

An = Pure premium

l = Premium loading factor.

3. Demography

i. Crude Death Rate =D

x1000P

Where,

D stands for total number of deaths in a given year, and

P stands for the size of the mid year population.

ii. Fertility Rates:

a. Crude Fertility Rate (CFR) =

Number of births during a specified

period

Total number of mid-year population

of women

b. General Fertility Rate (GFR) =

Number of births during a specified

period

Total number of mid-year population of

women aged

between 15-49

c. Age-Specific Fertility Rate,

at age y (ASFRy ) =

Number of births in a specified period to

women aged y years

Total number of mid-year population of

women aged y years

iii. Marriage Rates:

a. Crude Marriage Rate (CMR) =

Number of marriages taken place

during a specific period

Total number of mid-year population

b. General Marriage Rate (GMR) = 15

M

P+

x 1000

Where,

M stands for the total number of marriages solemnized in a given period and

P15+ stands for the mid-year population of age 15 years or more

ICFAI


9

c. Age-Specific Marriage Rate

at age y (ASMRy) =

Number of people married at age y during

the year

Total number of mid-year population at

age y

iv. Migration Rate of any area r =

Number of people moving in and out of

area r in a specified period

Total number of population in area r at

the beginning of the time period

v. Dependency ratio = Economically inactive population

Economically active population

4. Survival Models

i. The estimated probability of deaths in an interval computed per unit time,

iF = i i 1

i

P P

h

+−

Where,

iF = Respective probability density in the ith interval

iP = Estimated cumulative proportion surviving at the beginning of the ith

interval (at the end of the interval i – 1)

i 1P+

= Cumulative proportion surviving at the end of the ith interval

ih = Width of the ith interval

ii. Exponential Distribution

F(T) = Te−λλ =

( 1/ m)T1e

m

−

Where,

λ = Constant death rate in terms of deaths per unit of measurement

m = Mean time between deaths

T = Operating time, Life or age in hours, cycles, etc.

iii. Weibull Distribution

f(T) =

β 1β T

η η

−

β(T/η)e−

Where,

f(T) ≥ 0 ,T ≥ 0 ,β ≥ 0 and η > 0

η = Scale parameter

β = Shape parameter (or slope).

5. Mortality Tables

i. The probability that a person of age x years dies within one year

∴qx = Number of deaths between age x and x 1

Total number of persons living at age x

+= x x x 1

x x

d l l

l l

+−=

ICFAI

Formulae and Tables

10

ii. The probability that a person of age x years survives another one year

∴px = Number of survivors to age (x 1)

Total number of persons living at age x

+= x 1

x

l

l

+

iii. Expectation of life at age x is given by:

x 1

x

x

Ne

l

+′

∴ =

Where,

x 1

N+

′ = w 1

tt x 1

l−

= +

∑

w = Terminal age

iv. Central Death Rate: xx

x

2qm

2 q=

−

Where,

qx = The probability that a person of age x years dies within one year

v. The probability that a person of age x years survives another n years

npx = No. of persons living at age x n

No. of persons living at age x

+ = x n

x

l

l

+

vi. The probability that a person of age x years dies within the next n years

|mqx = Total no. of persons dying between ages x and x m

Total no. of persons living at age x

+ = x x m

x

l l

l

+−

vii. The probability that a person of age x years will die within n years following

m years from now m|nqx = No. of deaths between ages x m and x m n

No. of persons living at age x

+ + +

= x m x m n

x

l l

l

+ + +−

6. Assurance and Annuity Benefits

i. The present value of a term assurance of Re.1.00 payable on death during an n year

period is given by

1

x: nA =

x

1

l(vdx + v

2dx + 1 + v

3dx + 2 + ….. + v

ndx + n – 1)

Where,

x = Age of the person

n = Number of years the policy is in force


v = 1

l + i

dx = The number of deaths between age x and x + 1

lx = Total number of persons living at age x

ICFAI


11

ii. The present value of benefit of Re.1.00 payable to an insured against a pure

endowment policy for n years taken at an age x is given by:

x

1A

: n= v

n ×

x + n

x

l

l

Where,




v = 1

l + i


lx+n = Total number of persons living at age x + n

iii. The present value of benefit of Re.1.00 payable to an insured against an endowment

assurance policy for n years taken at an age x is given by:

x:n

A = 1

x:nA +

1

x:nA

Where,

1

x:nA = The present value of benefit in a Term Insurance Policy

1

x:nA = The present value of benefit in a Pure Endowment Policy

iv. The present value of an increasing whole life assurance on the life of a person aged

x at entry where the sum assured is Re.1.00 in the first year, Rs.2.00 in the second

year, Rs.3.00 in the third year and so on, is given by:

(IA)x = x

1

l(vdx + 2v

2dx+1 + 3v

3dx+2 + 4v

4dx+3 + ...)

Where,




v = 1

l + i

dx = The number of deaths between age x and x + 1


v. Commutation Functions:

a. Dx = vx lx

b. Cx = vx+1

dx

c. Mx = Cx + Cx+1 + Cx+2 + . . . . .

d. Rx = Mx + Rx+1

vi. Present value of the assurance benefits to the insured in terms of the commutation

functions are as follows:

a. Temporary Assurance Policy, 1

x:nA = x x+n

x

M M

D

−

b. Whole Life Assurance Policy, Ax = x

1

D(Mx)

ICFAI

Formulae and Tables

12

c. Pure Endowment Assurance Policy, 1

x:nA = x+n

x

D

D

d. Endowment Assurance Policy, x:n

A = x x+n x+n

x

M M + D

D

−

e. Double Endowment Assurance Policy:

DAx = x x+n x+n

x

M M + 2D

D

−

f. Increasing Temporary Assurance Policy:

1

x:n(IA) = x x+n x+n

x

R R nM

D

− −

g. Increasing Whole Life Assurance Policy, x(IA) = x

x

R

D

h. Special Endowment Assurance Policy that provides increasing death benefit

and increasing survival benefits:

x:n

(IA) = x x +n x +n x +n

x

R R nM + nD

D

− −

i. Deferred Temporary Assurance Policy:

1

x:nt A =

1

x t nA

: +–

1

x:tA

j. Deferred Whole Life Assurance Policy, xt A = Ax – 1

t:xA

vii. Present value of an immediate annuity for life of Re.1.00 to an annuitant of age

x years is given by ax = x 1

x

N

D

+

viii. Present value of an immediate annuity due for life of Re.1.00 to an annuitant of age

x years is given by xa&& = 1 + ax

Where,

ax = x+1

x

N

D

ix. Present value of a deferred life annuity for Re.1.00 to an annuitant of age x years for

a deferment period of t years is given by

x

t a = x t 1

x

N

D

+ +

x. Present value of a deferred life annuity for Re.1.00 due to an annuitant of age

x years for a deferment period of t years is given by:

x

t a&& = x t

x

N

D

+

xi. Present value of a temporary immediate life annuity for life of Re.1.00 to an

annuitant of age x years for a term of n years is given by

x:n

a = x 1 x n 1

x

N N

D

+ + +−

ICFAI


13

xii. Present value of a deferred temporary immediate life annuity for life of Re.1.00 to

an annuitant of age x years for a term of n years to be started after a deferment

period of t years is given by

x:n

t a&& = ax:n t 1+ −

– ax:t 1−

xiii. Present value of an increasing life annuity in terms of commutation function Sx is

given by:

x

(Ia)&& = x

x

S

D

xiv. Present value of an increasing life annuity in terms of commutation functions is

given by:

( )x:nIa = [ ]x x n x nx

1S S nN

D + +− −

xv. Present value of a life annuity with m number of payments in a year is given by:

(m)

x:na = x n

x:nm x

Dm 1a 1

2 D

+ +

+ −

7. Premiums for Assurance and Annuity Plans

i. The amount of level annual premium to be paid by a person of age x at the

beginning of each year to have a term assurance plan for n years:

1

x:nP =

x x n

x x n

M M

N N

+

+

−

−

ii. The amount of level annual premium to be paid by a person of age x at the

beginning of each year to have a pure endowment assurance plan for n years:

1

x:nP =

x n

x x n

D

N N

+

+−

iii. The amount of level annual premium to be paid by a person of age x at the

beginning of each year to have an endowment assurance plan for n years:

x:n

P = x x n x n

x x n

M M D

N N

+ +

+

− +

−=

1

x:nP +

1

x:nP

iv. The amount of level annual premium to be paid by a person of age x at the

beginning of each year to have a whole life assurance plan:

xx

x

MP

N=

v. The amount of level annual premium to be paid by a person of age x at the

beginning of each year to have a limited payment assurance plan for a limited period

of t years:

Pxt = x

x x t

M

N N+

−

vi. If the payment of premiums is limited to a shorter period ‘t’ where t < n years in an

endowment assurance plan then the level premium is denoted by Px:nt

Px:nt = x x n x t

x x t

M M D

N N

+ +

+

− +

−

ICFAI

Formulae and Tables

14

vii. The present value of a decreasing term assurance policy

A = x x 1 x n 1

x

S(nM R R )

D

+ + +− +

Where,

Sn = The amount of sum assured in the first year

S = The amount by which the amount of sum assured decreases in every

year

Rx, Mx and Dx are the different communication functions.

If ‘P’ is the net annual premium limited for a fixed period ‘t’ years

Where t ≤ 2n

3, then P =

( )x x 1 x n 1

x x t

S nM R R

N N

+ + +

+

− −

−

Where,

S = The amount by which, the amount of sum assured is reduced every

year

viii. Children’s Deferred Assurances:

a. Annual premium for the Children’s Deferred Whole Life Assurance plan is

given by:

21 x

21

21 x2121 x

v A

a v a

−

−

−

×

+ ×&& &&

Where, × is the age of the child.

b. Annual premium for the Children’s Deferred Endowment Assurance plan,

maturing at an age m is given by:

21 x

21:m 21

21 x

21 x 21:m 21

v A

a v a

−

−

−

− −

+&&

c. Additional Annual premium payable during the deferment period to get the

premium waiver benefit in the event of death of the father during the

deferment period, corresponding to the basic annual premium of Rs.100,

given by:

= ( )21 x

y:21 x

100 a

a

−

−

&&

&&

Where,

y = The age of the father on the date of commencement of the policy.

ix. Net single premium for an immediate annuity of Re.1.00 per annum payable in

arrear every year for n years certain and thereafter during the life time of the

annuitant of age x at entry is given by:

an

+ x nx n

x

Dx (a )

D

+

+

x. Net annual premium P payable for t years for the deferred annuity of Re.1 per

annum payable m times in a year for n years certain and thereafter during the

lifetime of the annuitant of age x at entry with a deferment period of t-years is given by:

P =

(m)

n

t

x n t

x n tx t

D m 1a a

D 2m

s

+ +

+ +

+

− + +

&&

ICFAI


15

xi. Calculation of premiums when frequency of payment is m times a year:

a. Let (m)xP represents the net premium per annum payable for a whole life

assurance at the end of the year of death of (x). A premium of (m)x

1P

m is

payable at the commencement of each m th period of a year which (x) enters.

(m) (m)x xP a&& = x

x

P

m 11 (P d)

2m

− − +

Where,

Px = The amount of annual premium

(m)xa&& = The present value of annuity due where the premiums are paid

m times a year

d = Discount factor = i

1 + i


b. For an endowment assurance Re.1 on (x) for a term of n years for which

premiums are payable m times, we have,

(m)

x:nP =

{ }

x:n

1

x:n

P

m 11 P d

2m

−− +

Where,

x:n

P = Level annual premium

c. For whole life limited payment policy we have,

tPx(m)

=

{ }

Px

1

x:t

t

m 11 P d

2m

−− +

Where,

Pxt = Level annual premium

d. For limited payment endowment policy :

(m)

x:nP =

( )

Px:n

1

x:t

t

m 11 P d

2m

−− +

Where,

Px:nt = Level annual premium

xii. Premiums for additional risks:

a. The sum assured is subject to an initial debt of tD that reduces by D every

year. The additional premium payable for an whole life assurance policy will

be: x x 1 x t 1

x xx

D(tM R R )P P

N

+ + +′ ′ ′− +

′ − =′

Where, xM′ , R′ and xN′ are the commutation functions for additional risks.

ICFAI

Formulae and Tables

16

b. The sum assured is subject to an initial debt of tD that reduces by D every

year. The additional premium payable for an endowment assurance policy

will be:

x:nP′ –

x:nP =

x x 1 x t 1

x x n

D(t M R R )

N N

+ + +

+

′ ′ ′− +

′ ′+

xiii. Calculation of Office premium:

a. Whole life assurance policy:

P1

( )

( )

2 2x 2

x

1 11

x

I KS P K

a

I K 1 K

a

− + +

=

−− −

&&

&&

Where,

P1 = Office premium

Px = Level annual premium

xa&& = The Present value of an immediate annuity due for life

of Re.1.00 to an annuitant of age x years

I1 and I2 = Initial expenses which are expressed per unit of

premium and per unit of sum assured respectively

K1 and K2 = Renewel expenses equal to which are expressed per unit

of premium and per unit of sum assumed respectively

b. Endowment Assurance Policy: P1

( )

( )

I K2 2

nx:n

I K1 1

n

2

1

S P Kx: a

1 Ka

x:

−

−

+ +

=

− −

&&

&&

Where,

P1 = Office premium

n

Px:

= Level annual premium

n

ax:

&& = The Present value of an immediate annuity due for life

of Re.1.00 to an annuitant of age x years for a term of

n years

I1 and I2 = Initial expenses which are expressed per unit of

premium and per unit of sum assured respectively

K1 and K2 = Renewel expenses equal to which are expressed per unit

of premium and per unit of sum assumed respectively.

8. Credibility Theory

i. When Normal approximation is applied to the Poisson distribution then, the

probability (P) that observation X is within ± k of the mean µ is given by:

P = 2 Ф(k n ) – 1

Where,

n = Number of claims

Ф stands for normal distribution

ICFAI


17

ii. The standard for full credibility for severity is given by

N = n02sCV

Where,

n0 = The full credibility standard for frequency

2sCV = The coefficient of variation for the claim size distribution

iii. Process variance for pure premium is given by:

Var (PP) = 2 2 2f S S fµ σ µ σ+

Where,

µf = Mean of the claim frequency distribution

µs = Mean of the claim severity distribution

2fσ = Variance of the claim frequency distribution

2Sσ = Variance of the claim severity distribution

iv. The expected number required for full credibility of pure premium

nF = n0(1 + 2SCV )

Where,

n0 = The ratio between the mean pure premium and the standard deviation

of pure premiums

CVS = The coefficient of severity

v. If the Poisson assumption does not hold good, general formula for the standard for

full credibility is given by:

nF = {y2/k

2}( 2 2 2

f f S sσ /µ σ /µ+ )

Where,

k = Allowance for the variance of the observed sampled frequency rate

y = Standard normal variation

µf = Mean of the claim frequency distribution

µs = Mean of the claim severity distribution

2fσ = Variance of the claim frequency distribution

2Sσ = Variance of the claim severity distribution

vi. B U&& HLAMANN Credibility is given by

Z = N

N k+

Where,

N is the number of observations and k is the B u&& hlamann credibility parameter.

9. Loss Distributions and Risk Models

i. Poisson Distribution:

a. P (N r)= = n re n

r 0, 1, 2,............r!

−

=

b. Mean = n

c. Variance = n

ii. Lognormal Distribution:

a. The PDF is defined as:

f(x) = 1

x 2πσ× exp

21 lnx µ

2 σ

− −

x > 0

ICFAI

Formulae and Tables

18

b. Mean = 21

exp2

µ + σ

c. Variance = exp (2µ + σ2) [exp (σ

2) –1]

iii. Pareto Distribution:

a. The PDF is defined as: f(x)

=

α 1α β

, x 0β + x

+

>

β

b. Mean of a pareto distribution is given by,

E(X) = β

1α −

c. Var (X) = 2

2

αβ

(α 2) ( 1)− α −

iv. Gamma Distribution:

a. The PDF is defined as: f(x) = βx α 1βe (x) 0 x

τ(α)

α

− −≤ <∞

b. Mean = α

β

c. Variance = 2

α

β

v. Individual Risk Model:

a. Expected Aggregate Loss: E(S)

= n

jj 1

E(Y )

=

∑ = n

j jj 1

q µ

=

∑

Where,

Yj = The amount of claim from the n-th policy

qj = The probability of a claim from the j-th policy

jµ = The amount of benefit associated with the j-th policy

b. Variance of aggregate loss: Var (S)

= n

j

j 1

Var(Y )=

∑ = 2

j j j

nq (1 q )µ

j 1

−∑=

Where,

Yj = The amount of claim from the n-th policy

qj = The probability of a claim from the j-th policy

jµ = The amount of benefit associated with the j-th policy

vi. Collective Risk Model:

a. Mean: E(S) = E[Yi] E[n]

b. Variance Var (S) = E[n] Var [Yi] + E[Yi] Var [n]

Where,

Yi = Amount of claim from the i-th policy

n = Number of policies.

ICFAI


19

10. Policy Values

i. For a Whole Life assurance policy, policy value is given by:

tVx = Ax+t, Px

Where,

Ax+t = Present value of Assurance benefits

Px = Level annual premium

ii. The policy value under prospective method for an Endowment assurance policy is

given by:

Vnx:t =

x t:n tA

+ − –

x:nP .

x t:n ta

+ −

&&

Where,

x t:n tA

+ −

= Present value of Assurance benefits an age of x + t years

x:nP = Level Annual Premium

x t :n ta

+ −

&& = Present value of an immediate annuity due for life of Re.1.00

to an annuitant of age x + t years for a term of n – t years.

iii. Under prospective method, the policy value for Temporary assurance policy is given

by:

1Vx:nt = 1

x t:n tA

+ −

– x t :n t

P+ −

.x t:n t

a+ −

&&

Where,

1

x t:n tA

+ −

= Present value of Assurance benefits an age of x + t years

x t :n tP

+ − = Level Annual Premium

x t :n ta

+ −

&& = Present value of an immediate annuity due for life for Re.1.00 to

an annuitant of age x + t years for a term of n – t years.

11. Surplus and it’s Distribution

i. Loading profit that is profit due to lower expenses is expressed as:

( P′ – P – E) x i

1+2

Where,

P′ = Total amount of office premium received

P = Total of premiums taken credit for in the last valuation

E = Actual expenses

i = Valuation rate.

ICFAI

II. Economics

1. Supply and Demand Analysis i. Price elasticity of demand

a. Point Elasticity

ep = Q P

P Q

∂×

∂

Where,

∂Q = Infinitisimal change in quantity demanded

∂P = Infinitisimal change in price

P = Original price of the good

Q = Original quantity demanded of the good

b. Arc Elasticity

ep = 0 1

0 1

P PQ

P Q Q

+∆×

∆ +

Where,

∆Q = Change in quantity demanded

∆P = Change in price of the good

P0 = Original price of the good

P1 = New price of the good

Q0 = Original quantity demanded of the good

Q1 = New quantity demanded of the good

ii. Marginal Revenue

MR = AR

p

11

e

−

Where,

AR = Average revenue

ep = Price elasticity of demand

iii. Income elasticity of demand

ey = Q Y

Y Q

∂×

∂

Where,

∂Q = Change in quantity demanded

∂Y = Change in income of the consumer

Y = Income of the consumer

Q = Quantity demanded of the good

iv. Cross price elasticity of demand

ecij = ji

j i

PQ

P Q

∂×

∂

Where,

∂Qi = Change in quantity demanded of the good i

∂Pj = Change in price of the good j

Pj = Price of the good j

Qi = Quantity demanded of the good i

ICFAI

Economics

21

v. Promotional elasticity of demand

eA = Q A

A Q

∂×

∂

Where,

∂Q = Change in quantity demanded

∂A = Change in units of advertisement expenditure on the good

A = Units of advertisement expenditure on the good

Q = Quantity demanded of the good

vi. Price-elasticity of supply

es = s

s

Q P

P Q

∂×

∂

Where,

P = Price of the good

Qs = Quantity supplied of the good

∂Qs = Change in quantity supplied on the good

∂P = Change in price of the good.

2. Consumer Behavior and Analysis

i. Marginal Rate of Substitution of good X for good Y

MRSX,Y = X

Y

MU

MU

Where,

MUX = Marginal Utility of good X

MUY = Marginal Utility of good Y

ii. Consumer equilibrium

X

X

MU

P = Y

Y

MU

P

Where,

MUX = Marginal Utility of good X

MUY = Marginal Utility of good Y

PX = Price of good X

PY = Price of good Y

iii. Budget constraint

I = PX X + PY Y

Where,

I = Income of the consumer

X = Number of units of good X

Y = Number of units of good Y

PX = Price of good X

PY = Price of good Y.

ICFAI

Formulae and Tables

22

3. Production Analysis

i. Average product of labor

APL = LTP

L

Where,

TPL = Total product of labor

L = Number of labor units

ii. Marginal product of labor

MPL = LTP

L

∆

∆

Where,

∆TPL = Change in total product of labor

∆L = Change in the number of labor units

iii. Marginal rate of technical substitution between Labor (L) and Capital (K)

MRTSL,K = L

K

MP

MP

Where,

MPL = Marginal product of labor

MPK = Marginal product of capital

iv. Cost constraint of a firm

C0 = wL + rK

Where,

C0 = A given amount of money that the firm spends

L = Number of labor units

K = Number of capital units

w = Wage rate

r = Interest rate

v. Efficient input combination

L

K

MP w

MP r=

Where,

MPL = Marginal product of labor

MPK = Marginal product of capital

w = Wage rate

r = Interest rate.

4. Analysis of Costs

i. TC = TFC + TVC

Where,

TC = Total cost

TFC = Total fixed cost

TVC = Total variable cost

ICFAI

Economics

23

ii. AFC = TFC

Q

Where,

TFC = Total fixed cost

Q = Number of units produced

iii. MC = TC

Q

∂

∂

Where,

∂TC = Change in total cost

∂Q = Change in quantity produced

iv. Break-even output (Q) = FC

P AVC−

Where,

P = Price

FC = Fixed cost

AVC = Average variable cost.

5. Market Structure: Perfect Competition

i. Profit of a firm (π) = TR – TC

Where,

TR = Total revenue

TC = Total cost

ii. Tax burden on the buyer = s

d s

e

e e+× Tax

Where,

es = Price elasticity of supply

ed = Price elasticity of demand

iii. Profit maximization

a. First order condition

MC = MR = AR = P

b. Second order condition

2 2

2 2

TR TC

Q Q

∂ ∂<

∂ ∂

Where,

TR = Total revenue

TC = Total cost

MR = Marginal revenue

MC = Marginal cost

AR = Average revenue

P = Price

Q = Quantity.

ICFAI

Formulae and Tables

24

6. Market Structure: Monopoly

i. Profit maximization

Marginal Revenue (MR) = Marginal Cost (MC)

ii. Lerner Index of monopoly power

a. L = P MC

P

−

b.

p

P MC 1

P e

−=

Where,

P = Price

MC = Marginal Cost

ep = Elasticity of demand

iii. The Herfindahl’s Index (H)

H = 2 2 2 21 2 3 nS S S ...S+ + +

Where,

S1 = % share of the largest firm in the market

S2 = % share of the second largest firm in the market

Sn = % share of the nth firm in the market.

7. Market Structure: Oligopoly

i. Output determination

Qn = P

nQ

n 1

+

Where,

Qp = Output if the market would be a competitive one

n = Number of firms in Oligopoly.

8. Measurement of Macro Economic Aggregates

i. Gross = Net + Depreciation

ii. Market Price = Factor Cost + [Indirect Tax – Subsidy]

iii. National = Domestic + Net Factor Income from Abroad

iv. The Laspeyre Price Index

It =

nt 0i i

i 1

n0 0i i

i 1

P q

100

P q

=

=

×

∑

∑

Where,

0iq = Quantity of ith good purchased in the base year

0ip = Price of the ith good in the base year

tiq = Quantity of ith good purchased in the current year

tip = Price of the ith good in the current year

v. GNP Deflator = NominalGNP

RealGNP.

ICFAI

Economics

25

9. The Simple Keynesian Model of Income Determination

i. Y = C + I + G + E – M

Where,

Y = Equilibrium income

C = Consumption expenditure

I = Investment expenditure

G = Government expenditure

E = Exports

M = Imports

ii. Average Propensity to Consume (APC) = C

Y

Where,


Y = Income

iii. Marginal Propensity to Consume (MPC) =C

Y

∆

∆

Where,

∆C = Change in consumption expenditure

∆Y = Change in income

iv. Average Propensity to Save (APS) = S

Y

Where,

S = Savings

Y = Income

v. Marginal Propensity to Save (MPS) = S

Y

∆

∆

Where,

∆S = Change in savings

∆Y = Change in income

vi. Multiplier (m) = [ ]

1

1 (1 t)− β − − π + µ

Where,

β = Marginal propensity to consume

t = Tax coefficient

π = Induced investment coefficient

µ = Marginal propensity to import.

10. Income Determination Model including Money and Interest

i. Goods market equilibrium

Y = C + I + G + X – M

Where,


Y = Income

ICFAI

Formulae and Tables

26

I = Investment expenditure

G = Government expenditure

E = Exports

M = Imports

ii. Money market equilibrium

Ms = Md

Where,

Ms = Supply of money

Md = Demand for money.

11. Money Supply and Banking System

i. High powered money (H) = Monetary liabilities of the Central bank +

Government money

ii. Multiplier (M) = u

u

1 C

C r

+

+

Where,

Cu = Currency deposit ratio

r = Cash reserve ratio

iii. Money supply (Ms) = H × m

Where,

H = High powered money

m = Money multiplier

iv. Finance Ratio = Total Issues

National Income

v. Financial Interrelation Ratio (FIR) = Total Issues

Net Capital Formation

vi. New Issue Ratio (NIR) = Primary Issues

Net CapitalFormation

vii. Intermediation Ratio (IR) = Secondary Issues

Primary Issues

viii. Velocity of money (v) = s

Y

M

Where,

Y = Income

Ms = Money supply.

12. The Open Economy and Balance of Payments

i. Trade balance = Exports – Imports

ii. Current account balance

= Credit (Current account) – Debit (Current account)

iii. Capital account balance

= Credit (Capital account) – Debit (Capital account).

ICFAI

Economics

27

13. Modern Macro Economics: Fiscal Policy, Budget Deficits and

Government Debt

i. Fiscal Deficit = Borrowings and other liabilities

ii. Primary Deficit = Fiscal deficit – Interest payments

iii. Revenue Deficit = Revenue expenditure – Revenue receipts.

ICFAI

III. Financial Management

1. Time Value of Money

i. Future Value of a Lump Sum (Single Flow)

FVn = PV(1 + k)n

Where,

FVn = Future value of the initial flow n years hence

PV = Initial cash flow

k = Annual rate of interest

n = Life of investment

ii. Effective rate of interest

r =

mk

1m

+

– 1

Where,

r = Effective rate of interest

k = Nominal rate of interest

m = Frequency of compounding per year

iii. Future Value Interest Factor of Annuity

FVIFA(k,n) = n(1 k) 1

k

+ −

Where,

k = Rate of interest

n = Time horizon

iv. Sinking Fund Factor = 1

FVIFA(k,n)

Where,

FVIFA(k,n) = Future value interest factor for annuity at k% for n years

v. Present Value Interest Factor of Annuity

PVIFA(k,n) = n

n

(1 k) 1

k(1 k)

+ −

+

Where,


n = Time horizon

vi. Capital Recovery Factor = 1

PVIFA(k, n)

Where,


n = Time horizon

vii. Present Value Interest Factor of a Perpetuity

P∞ = 1/k

Where,

k = Rate of interest.

ICFAI

Financial Management

29

2. Risk and Return

i. Rate of return

k = t t t 1

t 1

D (P P )

P

−

−

+ −

Where,

k = Rate of return

Pt = Price of security at time ‘t’, i.e., at the end of the holding period

Pt – 1 = Price of the security at time ‘t – 1’ i.e., at the beginning of the holding

period or purchase price

Dt = Income or cash flows receivable from the security at time ‘t’

ii. Expected rate of return ( )k = n

i i

i 1

p k=

∑

Where,

ki = Rate of return from the ith outcome

pi = Probability of the ith outcome

n = Number of possible outcomes

i = Outcome i

iii. Variance of an asset’s rate of return, VAR(k) = n

2i i

i 1

p (k k)=

−∑

Where,

VAR (k) = Variance of returns

pi = Probability associated with ith possible outcome

ki = Rate of return from the ith possible outcome

k = Expected rate of return

n = Number of years

i = Outcome i

iv. Standard deviation, σ = VAR(k)

v. CAPM model:

kj = rf + βj (km – rf)

Where,

kj = Expected or required rate of return on security ‘j’

rf = Risk-free rate of return

jβ = Beta coefficient of security ‘j’

km = Return on market portfolio

vi. Beta of security i, βi = im

2m

Cov

σ

Where,

Covim = Covariance of security i with true market

2mσ = Variance of returns on the market index

ICFAI

Formulae and Tables

30

vii. Alpha of security i (α) = E(ri) – R(ri)

= E(ri) – ( )( )f im m fr E r r + β −

Where,

α = The difference between expected return and required return

rf = Risk-free rate

βim = Beta coefficient of security i

E(ri) = Expected return of security i

R(ri) = Required return from security i

E(rm) = Return on market portfolio

viii. Systematic risk of security, i = 2 2im mβ σ

= 2 2 2im i m

2m

ρ σ σ

σ

= 2 2im iρ σ

= 2 2im iR σ

since 2 2im imR =ρ

Where,

βim = Beta coefficient of security i

2mσ = Market variance

2iσ = Variance of security i

2imρ = The correlation coefficient, and

2imR = The coefficient of determination between the security i and the market

portfolio

ix. Unsystematic Risk, ( 2eiσ ) = 2 2 2

i im mσ −β σ

Or

= 2 2 2i im iσ − ρ σ

= ( )2 2i im1σ − ρ

= ( )2 2i im1 Rσ −

Where,

2iσ = Variance of Security i

imβ = Beta coefficient of security i

2mσ = Market variance

2imρ = The correlation coefficient, and

2imR = The coefficient of determination between the security i and the market

portfolio.

ICFAI


31

3. Valuation of Securities

i. Equity Valuation:

a. The intrinsic value or present value of equity share

(P0) = ( )

nt n

t nt 1 ee

D P

(1 k )1 k=

+++

∑

Where,

P0 = Current market price of the equity share or intrinsic value of

the share

Dt = Expected equity dividend at time t

Pn = Expected price of the equity share at time n

ke = Expected rate of return or required rate of return

n = Investment period

t = Time t

b. The value of equity share when there is constant growth

P0 = 0

e

D (1 g)

k g

+

−

Where,

D0 = Current dividend per share

g = Expected constant growth rate in dividends


ii. Bond Valuation:

a. The intrinsic value or the present value of a bond

V0 or P0 = I(PVIFAkd, n) + F(PVIFkd, n)

Where,

V0 = Intrinsic value of the bond

P0 = Present value of the bond

I = Annual interest payable on the bond

F = Principal amount (par value) repayable at the maturity time

n = Maturity period of the bond

kd = Cost of Capital or Required rate of return

b. Current yield = Coupon Interest

Prevailing Market Price

c. Yield to maturity r in the equation

P0 = n

t nt 1

I F

(1 r) (1 r)=

++ +

∑

Where,




ICFAI

Formulae and Tables

32

iii. Valuation of a Convertible:

The value of convertible = n

n

t nt 1

(P ) Conversion ratioC

(1 r) (1 r)=

×+

+ +∑

Where,

C = Coupon amount

r = Required rate of return

Pn = Expected price of equity share on conversion

n = Number of years to maturity.

4. Financial Statement Analysis

i. Liquidity Ratios:

a. Current Ratio = Current Assets/Current Liabilities

b. Quick Ratio = Current Assets Inventories

Current Liabilities

−

c. Bank finance to working capital ratio = Short-term bank borrowings

Working capitalgap

ii. Leverage Ratios:

a. Long -term Debt-Equity Ratio = Long -termdebt

Net worth

b. Total Debt-Equity Ratio = Totaldebt

Net worth

c. Debt-Asset Ratio = Totaldebt

Total assets

iii. Coverage Ratios:

a. Interest coverage ratio = EBIT

Interest

Where,

EBIT = Earning before interest and tax

b. Cash flow coverage ratio = EBILT + D

LR PI L

(1 t) (1 t)+ + +

− −

Where,

EBILT = Earnings before interest, lease payments and taxes

D = Depreciation

I = Interest charges

L = Lease payments

t = Marginal Tax Rate

LR = Loan Repayment

P = Preference dividend

ICFAI


33

c. Debt Service Coverage Ratio

=

PAT Depreciation Other non-cash charges

Interest on term loan

Re payment of the term loanInterest on term loan

1 t

+ + + + −

Where,

PAT = Profit after tax

t = Marginal tax rate

iv. Turnover Ratios:

a. Inventory turnover = Cost of goodssold

Averageinventory

b. Accounts receivables turnover

= Net credit sales

Averageaccounts receivable

c. Total assets turnover = Net sales

Average total assets

v. Profitability Ratios:

a. Gross profit margin = Gross profit

Net sales

b. Net profit margin = Profit after tax

Net sales

c. Return on investment (Earning Power) = EBIT

Average total assets

Where,

EBIT = Earning before interest and tax

d. Return on Net Worth =Profit after tax

Average net worth

5. Financial Forecasting

i. External financing requirement

EFR = 1

A L( S) ( S) mS (1 d)

S S∆ − ∆ − −

Where,

EFR = External financing requirement

A/S = Current assets and fixed assets as a proportion of sales

∆S = Expected increase in sales

L/S = Spontaneous liabilities as a proportion of sales

m = Net profit margin

S1 = Projected sales for next year

d = Dividend pay-out ratio

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Formulae and Tables

34

ii. Sustainable growth rate (g) = 0

m(1 d)A / E

A / S m(1 d)A / E

−

− −

Where,

m = Net profit margin

d = Dividend pay-out ratio

g = Sustainable growth rate with internal equity

A/E = Total Assets

Equity = Current and fixed assets as proportion of equity

A/S0 = Current and fixed assets as proportion of sales at time 0.

6. Leverages

i. Degree of Operating Leverage (DOL) = [Q(S – V)] / [Q(S – V) – F]

Where,

Q = Quantity sold

S = Selling price per unit

V = Variable cost per unit

F = Total fixed deposit

ii. Degree of Financial Leverage (DFL) = p

EBIT

DEBIT I

(1 T)− −

−

Where,

I = Interest amount

Dp = Preference dividend

T = Tax rate

EBIT = Earnings Before Interest and Tax

iii. Degree of Total Leverage (DTL) = DOL × DFL

= p

Q(S V)

DQ(S V) F I

(1 T)

−

− − − −−

Where,

DOL = Degree of operating leverage

DFL = Degree of financial leverage

Q = Quantity sold




I = Interest amount


T = Tax rate

ICFAI


35

iv. Overall break-even point (Q) =

pDF I

(1 T)

(S V)

+ +−

−

Where,




I = Interest amount


T = Tax rate

v. Operating break-even point (Q) = F

(S V)−

Where,




vi. Financial break-even point (EBIT) = I + PD

(1 T)−

Where,

I = Interest amount


T = Tax rate.

7. Cost of Capital

i. Cost of Term Loans = I (1 – T)

Where,

I = Interest rate

T = Tax rate

ii. Cost of Debentures, n

t nt 1 d d

I(1 t) FP

(1 k ) (1 k )=

−= +

+ +∑

Where,

kd = Post-tax cost of debenture capital

I = Annual interest payment per debenture capital

t = Corporate tax rate

F = Redemption price per debenture

P = Net amount realized per debenture

n = Maturity period

iii. Cost of Preference Capital, n

t nt 1 p p

D FP

(1 k ) (1 k )=

= ++ +

∑

Where,

kp = Cost of preference capital

D = Preference dividend per share payable annually

F = Redemption price

ICFAI

Formulae and Tables

36

P = Net amount realized per share

n = Maturity period

iv. Cost of Equity Capital

a. Dividend forecast approach, Pe = 1

e

D

k g−

Where,

Pe = Price per equity share

D1 = Expected dividend per share at the end of one year

ke = Rate of return required by the equity shareholders

g = Growth rate of dividends

b. Cost of External Equity, 1e

o

Dk g

P (1 f )′ = +

− (Method 1)

ee

kk

(1 f )′ =

− (Method 2)

Where,

ek′ = Cost of external equity

ke = Cost of equity

D1 = Dividend expected at the end of year 1

Po = Current market price per share

g = Constant growth rate applicable to dividends

f = Floatation costs as a percentage of the current market price

v. Weighted Average Cost of Capital

= p deE P D

k k k (1 T)E P D E P D E P D

+ + −

+ + + + + +

Where,

E = Market value of equity

P = Market value of preference capital

D = Market value of debt

ke = Cost of equity

kp = Cost of preference capital

kd = Cost of debt

T = Tax rate.

8. Capital Structure

i. Overall capitalization rate of the firm

ko = kd B

B S+ + ke

S

B S+

Where,

kd = The cost of debt

B = The market value of the outstanding debt

S = The market value of equity

ke = The cost of equity

ko = The weighted average cost of capital

ICFAI


37

ii. Present value of a tax shield of interest payments:

a. When debt is perpetual = tcB

Where,

tc = The tax rate on corporate income

B = The market value of the debt

b. When corporate taxes are considered the value of the levered firm

V = cc

O(1 t )t B

k

−+

Where,

O = Operating income



k = Interest rate on debt

c. If the personal tax rate is tp, the tax advantage of debt = tc B (1 – tp)

Where,



d. When the tax rate on stock income (tps) differs from the tax rate on debt

income (tpd),

the tax advantage of debt capital = c ps

pd

(1 t )(1 t )1 B

(1 t )

− −− ×

−

Where,


B = The market value of the debt.

9. Dividend Policy

i. Traditional Model (Graham-Dodd Model), P = m (D + E/3)

Where,

P = The market price per share

m = The multiplier

D = The dividend per share

E = The earnings per share

ii. Walter Model, P = e

e

D (E D)r / k

k

+ −

Where,

P = The market price per share

D = The dividend per share

E = The earnings per share

r = The internal rate of return

ke = The cost of equity capital

iii. Gordon Model, P0 = 0

e

Y (1 b)

k br

−

−

Where,

P0 = The market price per share at the beginning of period 0

Y0 = The earnings per share for period 0

ICFAI

Formulae and Tables

38

b = The retention ratio (retained earnings/total earnings)

r = The return on investments

ke = The cost of equity capital or (Cost of capital of firm)

iv. MM Approach, P0 = 1 1

e

D P

1 k

+

+

Where,

P0 = The market price per share at the beginning of period 0

D1 = The expected dividend per share for period 1

P1 = The market price per share at the end of period 1

ke = The cost of equity capital

v. Corporate Dividend Behavior (Lintner Model)

Dt = cr EPSt + (1 – c) Dt – 1

Where,

Dt = The dividend per share for the time period t

c = The weightage given to current earnings by the firm

r = The target pay-out rate

EPSt = The earnings per share for the time period t

Dt – 1 = The dividend per share for the time period (t – 1).

10. Estimation of Working Capital Needs

i. Durations at various stages of production

a. Raw Material Storage Period =

AverageStock of Raw Material andStores

Average Raw Materials andStores consumed per day

b. Work-in-process period = Average Work-in-process inventory

Averagedailycost of production

c. Finished goods storage period = Average finished good inventory

Averagedaily cos t of sales

d. Average collection period = Average accounts receivable

Averagedailycredit sales

e. Average payment period = Averageaccounts payable

Averagecredit purchases per day

ii. Net operating cycle period = a + b + c + d – e

iii. Weighted Operating Cycle =

Dwoc = Wrm Drm + Wwip Dwip + Wfg Dfg + War Dar – Wap Dap

Dwoc = Duration of weighted operating cycle

Wrm = Weight of raw material expressed as a percentage of raw material cost

to sales

Drm = Duration of raw material

ICFAI


39

Wwip = Weight of work-in-progress expressed as a percentage of

work-in-progress cost to sales

Dwip = Duration of work-in-progress

Wfg = Weight of finished goods expressed as a percentage of cost of goods

sold to sales

Dfg = Duration of finished goods

War = Weight of accounts receivables expressed as a percentage of sales to

sales

Dar = Duration of accounts receivables

Wap = Weight of accounts payables expressed as a percentage of raw

material cost to sales

Dap = Duration of accounts payables.

11. Inventory Management

i. Economic Order Quantity

EOQ = 2UF

PCunits

Where,

U = Annual usage rate

F = Ordering cost

C = Carrying cost

P = Price per unit

ii. Reorder point = S × L + F (S R L)× ×

Where,

S = Usage in units

L = Lead time in days

R = Average number of units per order

F = Stock out acceptance factor.

12. Receivables Management

i. Effect of relaxing the credit standards on profit

∆P = ∆S(1 – V) – k ∆ I – bn ∆S

Where,

∆P = Change in profit

∆S = Increase in sales

V = Variable costs to sales ratio

k = Cost of capital

∆I = Increase in investment in receivables

= ∆S

360× Average collection period × V

bn = Bad debts loss ratio on new sales

1– V = Contribution to sales ratio

ICFAI

Formulae and Tables

40

ii. Effect of increasing the credit period on profit

∆P = ∆S (1 – V) – k ∆I – bn ∆S

The components of the formula are same excepting

∆I = (ACPN – ACPO) 0S

360

+ V(ACPN)S

360

∆

Where,


ACPN = New ACP (after increasing credit period)

ACPO = Old ACP

V = Ratio of variable cost to sales


k = Cost of capital

S0 = Sales before increasing the credit period

iii. The effect on profit for a change in cash discount rate

∆P = ∆S (1 – V) + k∆I – ∆ DIS

Where,


V = Ratio of variable cost to sales

k = Cost of capital

∆I = Savings in investment in receivables

= 0S

360(ACPO – ACPN) – V

S

360

∆ ACPN

∆DIS = Increase in discount cost

= pn (S0 + ∆S)dn – po S0 do

Where,

pn = Proportion of discount sales after liberalizing

So = Sales before liberalizing


dn = New discount percentage

p0 = Proportion of discount sales before liberalizing

d0 = Old discount percentage

ACPO = Average collection period before increasing cash discount

ACPN = Average collection period after increasing cash discount

iv. Effect of decreasing the rigor of collection program on profit:

∆P = ∆S(1 – V) – k∆I – ∆BD

Where,

∆P = Change in profits


V = Variable costs to sales ratio

k = Cost of capital

ICFAI


41


= oS

360(ACPN – ACPO) +

∆S

360ACPN × V

∆BD = Increase in bad debts cost

= bn (So + ∆S) – bo So

ACPO = Average collection period before relaxing collection effort

ACPN = Average collection period after relaxing collection effort

bo = Proportion of bad debts to sales before relaxing collection effort

bn = Proportion of bad debts to sales after relaxing collection effort.

13. Cash Management

i. Baumol Model, TC = I (C/2) + b (T/C)

Where,

TC = Total costs (total conversion costs + total holding costs)

I = Interest rate on marketable securities per planning period

C = Amount of securities liquidated per batch

T = Estimated cash requirement over the planning period

b = Fixed conversion cost

The point where total costs are minimum:

C = 2bT

I

ii. Miller and Orr Model, RP = 2

33b

4I

σ+ LL and,

UL = 3 RP – 2 LL

Where,

LL = Lower control limit

RP = Return point

UL = Upper control limit


I = Interest rate per day on marketable securities.

14. Capital Expenditure Decisions

i. Accounting Rate of Return

(ARR) = Average profit after tax

Average book valueof theinvestment

ii. Net Present Value (NPV)

NPV = n

t0t

t 1

CFI

(1 k)=

−+

∑

Where,

k = Cost of funds

CFt = Cash flows at the end of the period t

I0 = Initial investment

n = Life of the investment

ICFAI

Formulae and Tables

42

iii. Benefit-Cost Ratio (BCR)

BCR = PV

I

Where,

BCR = Benefit-Cost Ratio

PV = Present Value of future cash flows

I = Initial investment

iv. Net-benefit-cost Ratio

NBCR = NPV

I

Where,

NPV = Net present value


v. Internal Rate of Return (IRR)

I0 = n

t

tt 1

CF

(1 k)= +∑

Where,

k = IRR, which is that rate of return where n

t0t

t 1

CFI 0

(1 k)=

− =+

∑

CF = Cash flow

I0 = Initial investment

n = Life of investment.

ICFAI

IV. Financial Risk Management

1. Corporate Risk Management

i. Historical (ex-post)

a. Arithmetic mean return, n

i it

t 1

1r r

n=

= ∑

b. Variance (risk), ( )n

22i it i

t 1

1r r

n 1=

σ = −−∑

c. Standard deviation, i Varianceσ =

ii. Expected (ex-ante)

a. Expected return, E(ri) = n

is s

s 1

r P=

∑

b. Variance (Risk), [ ]n

22i is i s

s 1

r E (r ) .P=

σ = −∑

Where,

itr = Historical (ex post) return generated by the ith stock in time

period t

ris = Expected (ex ante) return for the ith stock assuming that S state

of the world occurs

Ps = Probability that the S state of the world will occur

ri = Return on a security ‘i’

iii. Estimated return on a stock (Rs) = α + β rm

Where,

rm = Return on market

β = Measure of stock’s sensitivity to the market index

α = Estimated return when the market return is zero

iv. According to the CAPM, the required return on a security

Rs = Rf + β(Rm – Rf)

Where,

Rf = Return on risk-free investment

Rm = Return on market

β = Measure of stock’s sensitivity to the market index.

2. Futures

i. Effective price = Sp2 + (Ft1 – Ft2)

If bases remains the same

Effective price = Sp1

Where,

Sp1 = Spot price at time t1

Sp2 = Spot price at time t2

Ft1 = Futures price at time t1

Ft2 = Futures price at time t2

Basis = Current cash price – Futures price

ICFAI

Formulae and Tables

44

ii. Margin

Initial margin = µ + 3σ

Where,

µ = Mean

σ = Standard Deviation

iii. Relationship between the cash price and the futures price of any commodity:

Ft,T = Ct + Ct × St,T × T t

365

−+ Gt,T

Where,

Ct = Cash price at time t

St,T = Annualized interest rate on borrowings

Gt,T = Storage costs

T – t = Time period

Ft,T = The futures price at time t, which is to be delivered at time period T

iv. Hedge Ratio (HR) = Futures position

Underlying asset position

v. Minimum variance hedge ratio, h = FpSp

Ft

σ

σ

Where,

h = Hedge ratio

Fp = Coefficient of correlation between Sp and Ft

σFt = Standard deviation of ∆ Ft

σSp = Standard deviation of ∆ Sp

∆Ft = Change of futures price during hedging

∆Sp = Change in spot price during hedging

vi. T-bill purchase price = Face value× %discount Days to maturity

1100 360

− ×

vii. IRR (Implied Repo Rate)

IRR = (FPt,T – CPt,T)/(CPt,T) × 360/T – t

Where,

FPt,T = Price of futures T-bill

CPt,T = Cash price of T-bill

T – t = Time period

viii. Transaction price or cash price of the bond,

P = Quoted price + Accrued interest

Invoice price = (Futures settlement price × Conversion factor) + Accrued interest

ix. HR = – Cash market principal

Conversion factorFutures market principal

×

ICFAI

Financial Risk Management

45

HR =

Cash flow to be hedgedConversion factor

Valueof futures contract

Portfolioduration

CTD bond duration

×

×

x. Change in value of a bond,

dB = – Duration

B dy1 y

× ×+

Where,

B = Value of the bond

y = Yield to maturity

dy = Change in yield

xi. Basis point value,

BPV = Duration

Market valueof bond 1bp(1 y / 2)

× ×+

HR = BPV (target) BPV (existing)

BPV (futures)

−

xii. Nf = – s T f

f s

DUR DUR 1 yS

DUR f 1 y

− + +

Where,

Nf = Number of futures contract required to change the duration to

DURT

DURs = Duration of bond with face value S

DURf = Duration of futures contract with price f

DURT = Target portfolio duration

yf = Yield implied by futures price

ys = Yield implied by spot portfolio

xiii. Treasury bond implied repo rate = ( )

1/(T t)

T T T

t t t

f CF AI1

f (CF ) AI

−

+ −

+

Where,

CFt = Conversion factor for bond delivered at t

CFT = Conversion factor for bond delivered at T

ft = Today’s futures price for contract expiring at t

fT = Today’s futures price for contract expiring at T

AIt = Accrued interest on bond as of time t

AIT = Accrued interest on bond as of time T

T – t = Time period.

3. Options

i. Pay-off from Buying a call option = Max (S – E, 0)

Pay-off from Buying a put option = Max (E – S, 0)

Where,

S = The market price of the underlying asset

E = The exercise price

ICFAI

Formulae and Tables

46

ii. Margin

a. Margin is higher of the following for naked out of the money option

Margin = Contract size × Option premium per share + 0.2 (Market value

of share) × Contract size – Contract size (amount by which

contract is out-of-the money)

Margin = Contract size × Option premium per share + 0.10 (stock’s

price) x Contract size

b. Margin for naked option (in-the-money)

= Contract size × Option premium per share + 0.20 (stock’s

market price) × Contract size

iii. Option price is a function of

Co or Po = f (So, E, σ2, t, rf, d)

Where,

Co = Value of call option

Po = Value of put option

f = Function of

E = Exercise price

So = Price of underlying stock

σ2 = Price volatility of underlying stock

t = Time to expiration

rf = Risk-free interest rate

d = Cash dividend

iv. Put-call parity equation

C + Xe–r(T–t)

= P + S

Where,

C = Call price

Xe–r(T–t)

= Present value of exercise price

P = Put price

S = Current market price

v. Binomial Pricing

Call price, C = u dC p C (1 p)

R

+ −

p = R d

u d

−

−

Where,

u = 1 + percentage increase in stock price from time 0 to time t

d = 1+ percentage decrease in stock price from time 0 to time t

C = The call price

Cu = The value of the call if the stock price increases

Cd = The value of call if the stock price decreases

R = 1+ risk-free rate of return (r)

p = Probability of price increase

vi. Black-Scholes option pricing model:

a. For a non-dividend paying stock

C = S0N(d1) – Xe– r(T – t)

N(d2)

ICFAI


47

P = Xe– r(T – t)

N(– d2) – S0N (– d1)

Where,

d1 =

2

02

In (S / X) r (T t)

(T t)

σ+ + −

σ −

d2 =

2

02

In (S / X) r (T t)

(T t)

σ+ − −

σ −

Or,

d2 = d1 – T tσ −

C = The call option price

P = The put option price

S0 = The spot price of the underlying asset

Xe–r(T–t)


r = The risk-free rate

(T – t) = The time to expiration expressed in years

σ = The annualized standard deviation of returns on the

underlying asset, i.e., the volatility measure

N(d) = Cumulative standard normal distribution

e = Exponential function

In = Natural logarithm

b. For a dividend paying stock:

C = S0 e–qt

N(d1) – Xe– rt

N(d2)

P = Xe– rt

N(– d2) – S0 e–qt

N (– d1)

Where,

d1 =

2

0

σIn (S /X) + r q + t

2

σ t

−

Or,

d2 = d1 – σ t

q = Dividend yield




Xe–r(T–t)



σ = The annualized standard deviation of returns on the

underlying asset, i.e., the volatility measure



In = Natural logarithm

ICFAI

Formulae and Tables

48

c. For a currency option:

C = S0 e– rf t

N(d1) – Xe– rt

N(d2)

P = Xe– rt

N(– d2) – S0 e– r

ft N (– d1)

Where,

d1 =

2

0 f

σIn (S /X) + r r + t

2

σ t

−



r = Domestic risk free rate

rf = Foreign risk free rate


X = The strike price of the option



In = Natural logarithm.

4. Swaps

i. Valuation of interest rate swaps, V = FB – FF

V = Value of the swap

FB = Value of fixed coupon bond

FF = Value of floating rate bond

ii. Valuation of currency swaps, V = PF – PL

V = Value of the swap

PF = Value of foreign currency bond

PL = Value of local currency bond.

5. Sensitivity of Option Premiums

i. Delta call = ∆C/∆S = N(d1)

Where,

∆C = Change in the call price

∆S = Change in the stock price

ii. Delta put = ∆C/∆S = N(d1) – 1

iii. Delta for portfolio of derivatives consisting of a single underlying asset:

∆P = n

j j

j 1

W=

∆∑

Where,

∆P = ∆ of portfolio

∆j = ∆ of j derivative

Wj = Weight of j derivative in the portfolio

iv. Theta of call = 1SN (d )

2 T t

′− σ

−– rXe

–r(T – t) N(d2)

S = The spot price of the underlying asset

ICFAI


49

1N (d )′ = 2 / 2

d1

1e

2

−

π

σ = The annualized standard deviation of returns on the underlying

asset, i.e., the volatility measure


v. Theta of put = 1SN (d )

2 T t

′− σ

−+ r Xe

–r(T – t) N(–d2)

Where,

d1 and d2 are defined as per Black-Scholes model.

N′ (d) = 2d / 2

1e

2

−

π

σ = The annualized standard deviation of returns on the underlying asset,

i.e., the volatility measure



vi. Vega of call or put = S T t− N′ (d1)

Where,

N′ (d1) =2 / 2

d1

1e

2

−

π



vii. Rho for a European put option = –X (T – t)e–r(T–t)

N(–d2)

viii. Rho for a European call option = X (T – t)e–r(T–t)

N(d2)

ix. Gamma of call or put = N′ (d1)/Sσ T t−

x. Portfolio Insurance:

Delta of a put on an index

∆ = e–q(T–t)

[N (d1) – 1]

d1 = 2In (S / X) (r q / 2) (T t)

T t

+ − + σ −

σ −

r = Domestic Risk-Free rate

q = Dividend yield.

6. Value at Risk

i. Daily volatility = Annual volatility

Number of working days

Daily value-at-Risk (VaR) (at a confidence level x%)

= Position Value x Daily Volatility × K(x)

Where,

K(x) = Factor relating to x% of confidence level.

ICFAI

V. International Finance

1. The Foreign Exchange Market

i. The conditions for no arbitrage possibility

a. (A/B)ask × (B/C)ask × (C/A)ask ≥ 1

b. (A/B)bid × (B/C)bid × (C/A)bid ≤ 1

ii. The annualized percentage premium on currency B for quote (A/B)

mid mid

mid

Forward(A / B) Spot(A / B) 12100

Spot(A / B) m

−× ×

Where,

m = Maturity of the forward contract in months.

2. Exchange Rate Determination

i. Interest rate parity (Investor’s decision)

a. Investment in currency A is profitable, if

( ) ( )A B

F(A / B)1 r 1 r

S(A / B)+ > × +

b. Investment in currency B is profitable, if

( ) ( )A B

F(A / B)1 r 1 r

S(A / B)+ < × +

c. The investor would be indifferent to the choice of currencies, if

( ) ( )A B

F(A / B)1 r 1 r

S(A / B)+ = × +

Where,

F(A/B) = Forward rate of currency B expressed in terms of

currency A

S(A/B) = Spot rate of currency B expressed in terms of

currency A

rA, rB = Investment rates in currencies A and B respectively

ii. Interest rate parity (Borrower’s decision)

a. Borrowing in currency A is profitable, if ( ) ( )A B

F(A / B)1 r 1 r

S(A / B)+ < × +

b. Borrowing in currency B is profitable, if ( ) ( )A B

F(A / B)1 r 1 r

S(A / B)+ > × +

c. The borrower would be indifferent to the choice of currency, if

( ) ( )A B

F(A / B)1 r 1 r

S(A / B)+ = × +

Where,

F(A/B) = Forward rate of currency B expressed in terms of

currency A

S(A/B) = Spot rate of currency B expressed in terms of

currency A

rA, rB = Borrowing interest rates in currencies A and B

respectively.

ICFAI

International Finance

51

3. International Project Appraisal

i. The adjusted present value of a foreign project

* * *nt t t

0 0 0 tt = 1 e

n n nt 0 t

0 0t tt = 1 t=1t=d b c

*n nt t

t tt = 1 t=1p i

(S C + E ) (1 T)APV = S (C A ) +

(1 + k )

D T rB T R+ + + S CL

(1 + k )t (1 + k ) (1 + k )1

P T I+ +

(1 + k ) (1 + k )

−− − ∑

−∑ ∑ ∑

∑ ∑

Where,

APV = Adjusted Present Value

S0 = Current exchange rate

C0 = Initial cash outlay in foreign currency terms

A0 = Activated funds

*tS = Expected exchange rate at time ‘t’

n = Life of the project

*tC = Expected cash flow at time ‘t’, in foreign currency terms

*tE = Expected effect on the cash flows of other divisions at time ‘t’, expressed

in domestic currency terms; can be either positive or negative

T = Domestic or foreign tax rate, whichever is higher

Dt = Depreciation in home currency terms at time ‘t’. (If the depreciation is

not allowed to be set off by the parent company against its own

profits, it needs to be defined in foreign currency terms with its

present value being converted at S0 into domestic currency terms)

B0 = Contribution of the project to borrowing capacity of the parent firm.

r = Domestic interest rate

CL0 = Amount of concessional loan received in foreign currency

Rt = Repayment of concessional loan at time ‘t’

*tP = Expected savings at time ‘t’ from inter-subsidiary transfer pricing

It = Illegally repatriated cash flows at time ‘t’

ke = All-equity discount rate, reflecting all systematic risks, including

country risk and exchange-rate risk

kd = Discount rate for depreciation allowances

kb = Discount rate for tax savings from generation of borrowing capacity

kc = Discount rate for tax savings due to concessionary loans, generally the

interest rate in the absence of concessionary loans

kp = Discount rate for savings through transfer pricing

ki = Discount rate for illegal transfers.

4. International Equity Investments

i. Variance of domestic currency returns on foreign investment

= Var(rf) + Var(S~) + 2 Cov(rf, S

~)

ICFAI

Formulae and Tables

52

ii. According to international CAPM, the return on a security

ri = rf + βw (rw – rf)

Where,

rf = World risk-free rate of return

βw = World beta of the security

= i w

w

Cov(r , r )

Var (r )

rw = Return on the world-market portfolio.

5. Short-term Financial Management

i. The break-even-size of investment

E = M[(k – i)/(k – d)]

Where,

E = Surplus funds at break-even level

M = Minimum lot of investment

k = Interest rate on borrowed funds

i = Rate of interest for investment

d = Rate of interest for deposit.

ICFAI

VI. Investment Banking and Financial Services

1. Money Market

i. Annual Turnover of Primary Dealer/Satellite Dealer

= Total Purchases andSales during the year

Average month-endstocks during the year

2. Rights Issues

i. Value of a Share after the Rights Issue = 0NP S

N 1

+

+

Where,

N = Number of existing shares for a rights share

P0 = Cum-rights market price per share

S = Subscription price at which the rights shares are issued

ii. Value of a Right (R) = rP S

N 1

−

+

Where,

R = Value of a right

Pr = Market value of share trading with rights

S = Strike price

N = Number of rights to purchase a new share

iii. Share Price Ex-Rights

a. Market Value of each Right after the Rights Issue, R = eP S

N

−

b. Value of Shareholding after Subscription = NP0 + S

Where,

Pe = Price of share ex-rights

S = Strike price

N = Number of rights to purchase a new share

P0 = Cum-rights market price per share.

3. Lease Evaluation

i. Lessee’s Angle

a. Weingartner’s Model:

∆NPV(L) = Initial Investment – P.V. (Lease Rentals) – Management

Fee + P.V. (Tax Shield on Lease Rentals) + P.V. (Tax

Shield on Management Fee) – P.V. (Tax Shield on

Depreciation) – P.V. (Net Salvage Value).

b. Equivalent Loan Model:

Net Value of Lease = + Initial Investment – P.V. (Lease Payment

Discounted at Kd) + P.V. (Tax Shield on Lease

Payments Discounted at kd) – P. V. (Depreciation

Tax Shield discounted at kd) – P.V. (Net Salvage

value Discounted at kd) – P.V. (Interest Tax Shield

on displaced Debt Discounted at kd)

Where,

Kd = Pre-tax marginal cost of debt

ICFAI

Formulae and Tables

54

kd = Post-tax marginal cost of debt

= Kd (1 – T)

T = Marginal tax rate

c. Bower-Herringer-Williamson (BHW) Model:

Financial Advantage of Leasing [FA(L)] =

Initial Investment – P.V. of Lease Payments

Or

FA (L) = P.V. of Loan Payments – P.V. of Lease Payments

Operating Advantage of Leasing [OA (L)] =

P.V. of lease related tax shield – P.V. of loan related tax shield – P.V. of

Residual Value

d. Bower’s Model:

Cost of Purchase (COP) = Initial investment – P.V. (Tax Shields on

depreciation discounted at an unspecified rate) –

P.V. (Net salvage discounted at marginal cost of

capital)

Cost of Leasing (COL) = P.V. (Lease Rentals discounted at pre-tax cost of

debt) – P.V. (Tax Shield on lease rentals

discounted at an unspecified rate) + P.V. (Tax

Shield on interest discounted at an unspecified

rate)

e. Suggested Framework for Lease Evaluation:

NAL = Investment Cost – P.V. (Lease Payments discounted at Kd) +

P.V. (Tax Shields on Lease Payments Discounted at k)–

Management Fee + P.V. (Tax Shield on Management Fee

discounted at k) – P.V. (Depreciation Tax Shields discounted at

k) – P.V.(Interest Tax Shields Discounted at k) – P.V.(Residual

value discounted at k)

ii. Lessor’s Angle

a. Present value of rental stream:

PV = L × (1 j)

(1 i)

+

+ + PVIFA(j,n)

Where,

PV = Present value of rental stream

Where, rentals increase/decrease at constant rate p.a.

L = Lease rental per period

n = Duration of lease in years

j = [(i – g)/(1 + g)]

i = Pre-tax yield p.a.

g = Constant rate of increase/decrease p.a.

b. Net Advantage of Leasing (NAL):

NAL = – Initial Investment + P.V. (Lease Payments) – P.V. (Tax

Lease Payments) + P.V. (Management Fee) – P.V. (Tax on

Management Fee) + PV. (Tax shields on Depreciation) + P.V.

(Net Salvage Value) – P.V. (Initial Direct Costs) + P.V. (Tax

Shield on Initial Direct Costs)

ICFAI

Investment Banking and Financial Services

55

c. Gross Yield:

The gross yield of a lease can be defined as that compounded rate of return

(discount rate) that equates: P.V (Lease Rentals) + P.V (Residual Value) to

Investment cost

Where, management fee and initial direct costs are involved the gross yield

will be the discount rate that equates:

P.V. (Lease Rentals) + P.V. (Residual Value) + Management Fees

= Investment Cost + Initial Direct Costs)

d. Add on yield = Annualcharge for credit

× 100Initial investment

e. IRR based pricing:

i = iF + ie + id

Where,

i = Risk adjusted rate of return

iF = Risk-free rate of return

ie = Premium for the risk characterizing the existing lease

investments

id = Premium for the differential risk characterizing the lease

investment under review

f. Value of the asset or the implied interest return earned by the lessor

= ( ) ( )

mn

t mnt 1

Lease Payments Lease Value

1 R / n 1 R / n=

+

+ +∑

Where,

n = Length of the lease term

m = Number of lease payments in a year

R = Implied Interest Return

If the lease payments are made in advance, mn mn 1

t 1 t 0

would bechanged to−

= =

∑ ∑

g. Internal Rate of Return or After tax cost of leasing

= n

t t t

t t nt 1

L T(L D ) RVA

(1 r) (1 r) (1 r)=

−− + + −

+ + +∑

Where,

A = The cost of the asset to be leased

Lt = The periodic lease payments at the end of the each period

T = The corporate tax rate

n = The lease term

Dt = The depreciation that can be claimed for tax purpose

RV = The residual value of the asset.

4. Hire Purchase

i. From Hirer’s Angle:

a. COHP = Down payment + P.V (Hire Payments) + Service Fee – P.V

(Tax shields on charge for credit of Hire payments & Service

Fee) – P.V (Tax shields on Depreciation) – P.V (Net salvage

value)

ICFAI

Formulae and Tables

56

b. COL = P.V (Lease payments) + Lease management fee – P.V (Tax

shields on lease payments & lease management Fee)

ii. From Finance Company’s Angle:

a. NPV (Lease Plan) = – Initial Investment – Initial Direct costs + P.V

(Lease Rentals) + Lease Management Fee + P.V

(Tax shields on Initial direct costs &

Depreciation) + P.V (Net Salvage Value) – P.V

(Tax liability on Lease Rentals and Lease

Management Fee)

b. NPV (HP Plan) = – Loan amount – Initial Direct costs +

Documentation & Service Fee + P. V (H.P

installments) – P.V (Interest Tax on Finance

Income) – P.V (Income Tax on Finance Income

netted for interest tax) + P.V (Tax shield on

initial Direct costs) – P.V (Income Tax on

Documentation & Service Fee)

iii. Effective rate of interest:

If payments are made in arrears,

I(app) = n

2Fn 1

×+

If the payments are made in advance,

iapp = n

2Fn 1

×−

Where,

F = Flat rate of interest per unit time

N = Total number of repayments

iv. Interest Rebate:

Rule of 78 method

R = ( )

( )

t t 1D

n n 1

+×

+

Where,

t = Number of level installment that are not due and outstanding

n = Total number of level installment

D = Total change for credit

R = Interest rebate

Under modified Rule of 78,

Interest Rebate = ( )( )

( )

t t 1D

n n 1

− α − α +×

+

Where,

α = Deferent period

Under Hire Purchase Act, 1972,

Interest rebate = 2 t

D3 n

× ×

5. Consumer Credit

i. The effective rate of interest is the discount rate in the equation:

Loan Amount – P.V (Installments paid) – Service Fee + P.V (Accumulated Value of

Deposit) + P.V (Prompt Payment Bonus) = 0

ICFAI

Investment Banking and Financial Services

57

6. Housing Finance

i. Disbursement Amount, RD

= AV × CC PC LC

AV BC CM100 100 100

× + × − −

Where,

RD = Recommendation for disbursement in rupees

AV = Aggregate value = LC + CC

PC = Progress of construction in % points

LC = Land component

CC = Cost of construction + Overheads + Profits

BC = Borrower’s contribution

CM = Cumulative disbursement made

ii. Equated Monthly Installments = n

n

1 Lr(1 r)

12 (1 r) 1

+

+ −

Where,

L = Loan

r = Rate of interest in decimals

n = Period.

7. Venture Capital

i. NPV = [(Cash) / (Post)] × [(PAT × PER)] × k

Where,

NPV = Net Present Value of the cash flows relating to the investment

Post = Pre + cash

Cash represents the amount of cash.

‘pre’ = The pre-money valuation of the firm estimated by the

‘investor’

k = The PVIF for the investment horizon

PER = Price Earnings Multiple

PAT = Profit After Tax.

ICFAI

VII. Management Accounting

1. Cost-Volume-Profit Analysis

i. Break-Even Point (Units) = Fixed Cost

Selling Price per Unit VariableCost per Unit−

= Fixed Cost

Contribution per Unit

Or, Break Even Sales(Rs.)

Selling Price per Unit

ii. Break-Even Point (Rs.)

= Fixed Cost xSelling Price per Unit

Selling Price per Unit VariableCost per Unit−

= Fixed Cost x Selling Price per Unit

Contribution per Unit

= Fixed Cost

Contribution per unit ÷ Selling price per unit

= Fixed Cost

P / V ratio =

Fixed Cost

Variable Cost1

Sales−

Or,

Break-even Point (Units) × Selling Price per Unit

iii. At Break-even Point

Sales – Variable Cost – Fixed Cost = 0

Or, Contribution – Fixed Cost = 0

Or, Contribution = Fixed Cost

iv. Calculation of Required Sales value to earn a desired amount of profit

= Fixed Cost + Desired Profit

P/V Ratio

v. Profit/Volume Ratio

a. P/V Ratio = Sales VariableCost

x100Sales

−

= Contribution

x100Sales

= Fixed Cost Pr ofit

x100Sales

+

Or,

Selling price per unit Variablecost per unit

x 100Selling price per unit

−

= Contribution per unit

x 100Selling price per unit

ICFAI

Management Accounting

59

b. P/V ratio = Change in Contribution

x100Change in Sales

Or

Change in Contribution per unit

x 100Change inSelling Price per unit

Or

Change in Profit

x100ChangeinSales

vi. Margin of Safety = Total Sales – Break-even Sales

Or Profit

P / V ratio

= Profit Selling price per unit

Selling price per unit Variablecost per unit

×

−

vii. Margin of Safety as a percentage of Total Sales

= Margin of Safety

x 100TotalSales

2. Standard Costing and Variance Analysis

i. Material Cost Variance

= Usage Variance + Price Variance

ii. Material Cost Variance = (SQ × SP) – (AQ × AP)

iii. Material Usage Variance = (SQ – AQ) × SP

iv. Material Price Variance = (SP – AP) × AQ

Where,

SQ = Standard Quantity for the actual output

SP = Standard Price

AQ = Actual Quantity

AP = Actual Price

v. Material Mix Variance = (Standard cost of standard mix of the actual

quantity – Standard cost of actual mix of the

actual quantity)

Or

= (Revised standard mix of actual input – Actual

mix) × Standard Price

vi. Material Yield Variance = (Standard yield specified – Actual yield) ×

Standard cost per unit

Or

= (Standard loss on actual input – Actual loss) ×

Standard cost per unit

vii. Sub-usage Variance = (Standard quantity – Revised standard proportion

of actual input) × Standard cost perunit of input

viii. Labor Cost Variance = Efficiency Variance + Rate Variance

ICFAI

Formulae and Tables

60

ix. Labor Cost Variance = (SH × SR) – (AH × AR)

Where,

SH = Standard Hours

SR = Standard Rate

AH = Actual Hours

AR = Actual Rate

x. Labor Efficiency Variance = (Standard hours for the actual output –

Actual hours) × Standard rate per hour

xi. Labor Rate Variance = (Standard rate – Actual rate) × Actual

hours

xii. Labor Mix Variance = (Revised standard labor mix in terms of

actual total hours – Actual labor mix) ×

Standard rate per hour

xiii. Labor Yield Variance = (Standard output based on actual hours –

Actual output) × Average standard labor

rate per unit of output

Or

= (Standard loss on actual hours – Actual

loss) × Average standard labor rate per

unit of output

xiv. Labor Efficiency Sub-variance = (Standard mix – Revised Standard mix) ×

Standard rate.

ICFAI

VIII. Portfolio Management

1. Capital Market Theory

i. Variance of a portfolio of n securities: n n

2n i j ij

i 1 j 1

W W= =

σ = σ∑∑

Where,

Wi = Weight of ith security

Wj = Weight of jth security

ijσ = Covariance between ith security and jth security

When portfolios are equally weighted,

the expected level of portfolio risk can be expressed as

( ) ( ) ( ) ( )2 2n i ij ijE 1/ n E E E σ = σ − σ + σ

Where,

( )2iE σ = Average variance of an individual security that is included in the

portfolio

( )ijE σ = Average pair wise covariance between securities in the portfolio

ii. Tax-adjusted CAPM E(ri)

= rf (1 – T) +βi [E(rM) – rf (1 – T) – TDm] + TDi

Where,

E(ri) = Expected return on stock i

rf = Risk-free rate of interest

βi = Beta coefficient of stock i

Dm = Dividend yield on the market portfolio

Di = Dividend yield on stock i

T = ( )

( )

d g

g

T T

1 T

−

−

= Tax factor

Td = Tax rate on dividends

Tg = Tax rate on (long-term) capital gains.

2. Arbitrage Pricing Theory (APT Model)

i. E(ri) = 0 1 i1 2 i2 3 i3 M iM...τ + τ β + τ β + τ β + + τ β

Where,

E(ri) = Expected return on Asset i

τ0 = Expected return on an asset with zero systematic risk

= rf if riskless borrowing and lending exist

τj = Risk premium, or market price of risk, associated with the jth factor

= E(rj) – τ0, or τj riskless borrowing and lending exist

βij = Sensitivity or beta coefficient for security i that is associated with index j.

ICFAI

Formulae and Tables

62

3. Asset Allocation

i. Umk = E (Rm) – 2m

kt

σ

Where,

Umk = The expected utility of asset mix m for investor k

E(Rm) = The expected return for asset mix m

2mσ = The standard deviation for asset mix m

tk = Investor k’s risk tolerance.

4. Delineating Efficient Frontiers

i. Optimal portfolio selection using sharpe’s optimization

a. Cut-off point (Ci) =

i2 i fM i2

i 1 ei

2i2 iM 2

i 1 ei

(R R )

1

=

=

−σ β

σ

β+ σ

σ

∑

∑

b. The proportion invested in each security, ii N

j

j 1

ZX

Z=

=

∑

c. The relative investment in each security

Zi = *i i f

2iei

R RC

β − −

βσ

Where,

2Mσ = Variance in the market index

2eiσ = The stock unsystematic risk

Ri = Expected return on stock i

Rf = Risk-free rate of return

β = Beta of the stock.

5. Portfolio Analysis

i. Expected return of a portfolio of n securities, n

p i i

i 1

E W E (R )=

=∑

Where,

Ep = The portfolio return

Wi = The proportion of investment in security i

E(Ri) = The expected return on security i

n = The total number of securities in the portfolio

ii. Holding period yield = ( )it it 1 t

it 1

P P D

P

−

−

− +

Where,

Pit = The current price of the security

Pit – 1 = The price of the security at the beginning of period t

Dt = The dividend received during the period t

ICFAI

Portfolio Management

63

iii. Variance or Risk of a portfolio

Var(Rp) = n n n

2i i i j i j

i 1 j 1 i 1, i j

W Var(R ) W W Cov(R R )= = = ≠

+∑ ∑ ∑

Where,

Var(Rp) = The variance of the return on the Portfolio

Var(Ri) = Variance of return on security i

Cov(Ri Rj) = The covariance between the returns of securities i and j

Wi, Wj = The percentage of investable funds invested in securities i and j.

iv. Correlation Co-efficient, ij

ij

i j

σρ =

σ σ

Where,

σij = Covariance between securities i and j

σi = Standard deviation of security i

σj = Standard deviation of security j

v. Systematic risk of security i = 2 2im mβ σ

Where,

2imβ = The beta of the security i

2mσ = The variance of the market portfolio

vi. Systematic risk of the portfolio =

2n

2i im m

i 1

X=

β σ

∑

Where,

Xi = Proportion of the total portfolio invested in security i

n = Total number of stocks

2imβ = The beta of the security i

2mσ = The variance of the market portfolio

vii. Unsystematic risk of portfolio = n

2 2i ei

i 1

X=

σ∑

Where,


n = Total number of stocks

2eiσ = Variance in security not caused by its relationship to the index

viii. Total portfolio variance (risk), 2pσ =

2n n

2 2 2i im m i ei

i 1 i 1

X X= =

β σ + σ

∑ ∑

Where,

2pσ = Variance of portfolio return

2mσ = Expected variance of index

2eiσ = Variance in security not caused by its relationship to the index


n = Total number of stocks.

ICFAI

Formulae and Tables

64

6. Portfolio Performance

i. Jensen’s differential return (αi) = Ri – [Rf + βi (Rm – Rf)

Where,

Ri = Average realized return on portfolio P

Rf = Risk-free return for period t

Rm = Average return of the market portfolio for period t

βi = A measure of systematic or market risk.(slope of the regression

equation)

αi = Intercept that measures the forecasting ability of the portfolio manager

ii. Treynor’s ratio = P f

p

(R R )−

β

Where,

PR = Return on the portfolio

fR = Risk-free rate of return

pβ = Beta of the portfolio

iii. Sharpe’s ratio = P f

P

R R−

σ

Where,

PR = Return on the portfolio

fR = Risk-free rate of return

Pσ = Standard deviation of return on the portfolio

iv. Return from total selectivity

= Return from net selectivity + Return for extra diversifiable risk

(or)

Return from net selectivity

= Return from total selectivity – Return for extra diversifiable risk

v. Return from net selectivity = ( ) PP f m f

m

R R R R σ

− + − σ

Where,

Rp = Return on portfolio

σp = Standard deviation of returns of portfolio p

σm = Standard deviation of market returns

Rf = Risk-free rate

Rm = Return on market portfolio.

ICFAI

Portfolio Management

65

7. Bond Portfolio Management

i. Number of futures contracts, X = T I I

F F

(D D ) P

D P

−

Where,

X = Approximate number of futures contracts

DT = Target effective duration for the portfolio

DI = Initial effective duration for the portfolio

DF = Effective duration for the futures contract

PI = Initial market value of the portfolio

PF = Market value of the futures contract.

ICFAI

IX. Project Management

1. Appraisal Criteria

i. Cash flow as per long-term funds point of view

= PAT + Depreciation + Interest on long-term (1 – t)

ii. Cash flow as per equity funds point of view

= PAT + Depreciation – Repayment of long-term borrowings – Repayment of

short-term bank borrowings

iii. Modified Net Present Value

NPVn = n

TVI

(1 k)−

+

Where,

NPVn = Modified net present value

TV = Terminal value

k = Cost of capital

I = Investment outlay

TV = n

n tt

t=1

CF (1+ r )−

′∑

n = Project life

Where,

CFt = Cash in flow at the end of the year t

r′ = Reinvestment rate applicable to the cash inflows of the project

iv. Modified Internal Rate of Return

r* =

1/ nTV

II

−

Where,


r* = Modified IRR

n = Project life

TV = Terminal value

I (1 + r*)

n = TV.

2. Risk Analysis in Capital Investment Decisions

i. Expected NPV and Standard Deviation of NPV

a. In perfectly correlated cash flows

Expected NPV ( )NPV = n

t

t 1

A=

∑ /(1 + i) t – I

S.D. of the NPV = n

t

t 1=

σ∑ /(1 + i) t

b. In uncorrelated cash flows

Expected NPV ( )NPV = n

t

t 1

A=

∑ /(1 + i)t – I

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Project Management

67

S.D. of the NPV =

1/ 2n

2 2tt

t 1

/(1 i)=

σ +

∑

Where,

tA = The expected cash flows for a time period t

i = The risk-free discount rate

n = The life of the project

NPV = The expected net present value

tσ = Standard deviation of the cash flows for a time period t

I = Initial investment.

3. Application of Portfolio Theories in Investment Risk Appraisal

i. Asset beta, βA = E D

E Dβ

E D E D

β +

+ +

Where,

βA = Asset beta

βE = Equity beta

βD = Debt beta.

4. Social Cost Benefit Analysis

i. Effective Rate of Protection

= Valueadded at domestic prices Valueadded at world prices

Valueadded at world prices

−

ii. Domestic resource cost = Valueadded at domestic prices Exchange rate

Valueadded at world prices

×

5. Options in Investment Appraisal

i. Transaction price or cash price of the bond,

P = Quoted price + Accrued interest

Invoice price = (Futures settlement price × Conversion factor) + Accrued interest

ii. HR = – Cash market principal

Conversion factorFutures market principal

×

iii. HR = Cash flow to be hedged Portfolio duration

Conversion factorValueof futures contract CTD bond duration

× ×

iv. Binomial Pricing Model

Call price, C = u dC p C (1 p)

R

+ −

p = R d

u d

−

−

Where,

u = 1 + percentage increase in stock price from time 0 to time t

d = 1+ percentage decrease in stock price from time 0 to time t

C = The call price

Cu = The value of the call if the stock price increases

ICFAI

Formulae and Tables

68

Cd = The value of call if the stock price decreases

R = 1+ risk-free rate of return (r)

p = Probability of price increase

v. Black-Scholes option pricing model:

C = S0N(d1) – Xe– r(T – t)

N(d2)

P = Xe– r(T – t)

N(– d2) – S0N (– d1)

Where,

d1 =

2

02

In (S / X) r (T t)

(T t)

σ+ + −

σ −

d2 =

2

0In (S / X) r (T t)2

(T t)

σ+ − −

σ −

Or,

d2 = d1 – T tσ −




X = The strike price of the option



σ = The annualized standard deviation of returns on the underlying asset,

i.e., the volatility measure



In = Natural logarithm.

6. Project Scheduling

i. Expected time (te) = o m pt 4t t

6

+ +

ii. Variance (V) =

2

p ot t

6

−

Where,

to = Optimistic estimate of time

tm = Most likely time

tp = Pessimistic estimate of time.

7. Project Monitoring and Control

i. Cost Performance Index = BCWP

ACWP

ii. Schedule Performance Index = BCWP

BCWS

ICFAI

Project Management

69

iii. Estimated Cost Performance Index = BCTW

ACWP ACC+

Where,

BCWP = Budgeted cost of work performed

ACWP = Actual cost of work performed

BCWS = Budgeted cost of work scheduled

BCTW = Budgeted cost for total work

ACC = Additional cost for completion.

ICFAI

X. Quantitative Methods

1. Basics of Mathematics

i. Progressions

a. The nth term in an A.P.

Tn = a + (n – 1)d

b. Sum of all the terms in an A.P.

S = n

2{2a + (n – 1)d}

Where,

a = First term

n = No. of terms

d = Common difference

c. The nth term in a G.P.

Tn = arn–1

d. Sum of numbers in a G.P.

S = ( )

( )

na r 1

r 1

−

− r ≠ 1

e. Sum of an Infinite G. P.

S = a

1 r−

Where,

a = First term

r = Common ratio

ii. Permutations and Combinations

a. Permutations

nPr =

( )

n!

n r !−

Where,

n! = n(n – 1) (n – 2) (n – 3) ...... 3.2.1

b. Combinations

nCr =

( )

nrP n!

r! n r !r!=

−

iii. Logarithms

a. loga MN = loga M + loga N

b. loga (M/N) = loga M – loga N

c. loga (pM ) = p. logaM

d. logba × logab = 1

2. Calculus

i. Rules of Differentiation

a. If f(x) = xn

then f ′ (x) = nxn – 1

b. If f(x) = g(x) h(x)

then f ′ (x) = g′ (x) h(x) + g(x) h′ (x)

ICFAI

Quantitative Methods

71

c. If f(x) = ( )

( )

g x

h x where h(x) ≠ 0

then f ′ (x) =

( )2

g (x)h(x) g(x)h (x)

h x

′ ′−

d. If f(x) = c. g(x) where ‘c’ is a constant,

then f (x) cg (x)′ ′=

e. If f(x) = g(x) + h(x)

then f ′ (x) = g′ (x) + h′ (x)

f. If f(x) = ln x, then f ′ (x) = 1

x

If f(x) = e g(x)

then f ′ (x) = g′ (x).eg(x)

g. If f(x) = ln g(x),

then f ′ (x) = ( )

( )

g x

g x

′

h. fn(x) =

n

n

d f (x)

dx

ii. Partial Derivatives

a. For a function, f = g(x,y) . h(x,y)

f

x

∂

∂ = g (x,y)

h

x

∂

∂ + h (x,y)

g

x

∂

∂

f

y

∂

∂ = g (x,y)

h

y

∂

∂ + h (x,y)

g

y

∂

∂

b. For a function, f = ( )

( )

g x, y

h x, yand h (x,y) ≠0,

[ ] [ ]

( )2

h(x, y) g / x g(x, y) h / xf

x h x, y

∂ ∂ − ∂ ∂∂=

∂

[ ] [ ]

( )2

h(x, y) g / y g(x, y) h / yf

y h x, y

∂ ∂ − ∂ ∂∂=

∂

c. For a function, f = [g (x,y)] n

[ ]n 1f g

n g(x, y)x x

−∂ ∂=

∂ ∂

[ ]n 1f g

n g(x, y)y y

−∂ ∂=

∂ ∂

iii. Integration

a. If ‘K’ and ‘c’ are constants, then

Kdx∫ = Kx + c

b. n n 11

x dx x c,n 1n 1

+= + ≠ −

+∫

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72

c. 1

x dx ln x c, x 0−

= + >∫

d. Kx

Kx aa .dx c

K ln a= +∫ , where ‘a’ and ‘K’ are constants

e. Kx

Kx ee .dx c

K= +∫

f. ( )Kf (x)dx K f x dx=∫ ∫

g [ ]f (x) g(x) dx f (x)dx g(x)dx+ = +∫ ∫ ∫

h. ( ) ( )[ f x dx] [f x dx]− = −∫ ∫

iv. Definite Integral

a. ( )

bb

aa

f (x)dx F x= ∫ = F(b) – F(a)

Where, F(x) is the indefinite integral of f(x)

b.

d c

c d

f (x)dx f (x)dx= −∫ ∫

c.

k

k

f (x)dx F(k) F(k) 0= − =∫

d.

qr r

p p q

f (x)dx f (x)dx f (x)dx= +∫ ∫ ∫

Where, p ≤ q ≤ r

e.

d d

c c

f (x)dx g(x)dx±∫ ∫ = ( )

d

c

f x g(x) dx± ∫

f.

r r

q q

cf (x)dx c f (x)dx=∫ ∫

Where, c is a constant.

3. Interpolation and Extrapolation

i. Linear Approximation Method of Interpolation

Interpolated figure

a. For ascending series:

= Base value + ( )i p

s p

Upper limit Lower limitt t

(t t )

−× −

−

b. For descending series:

= Base value – ( )i p

s p

Lower limit Upper limitt t

(t t )

−× −

−

Where,

Base value is the value of the immediately preceding year.

(ti – tp): time interval between the immediately preceding year and the year

for which the value is to be interpolated.

(ts – tp): time interval between the two known values.

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4. Central Tendency and Dispersion

i. Arithmetic Mean

( )

n

i

i 11 2 n

x

x x x ........ x / nn

=

= + + + =

∑

Where,

n = No. of observations

ii. Mean for discrete series or ungrouped data

k

i i

i 1

n

i

i 1

f x

x

f

=

=

=

∑

∑

Where,

f = Frequency

iii. Mean for continuous series or grouped data, fm

xN

∑=

Where,

m = Midpoint of class

= Lower limit + Lower limit of next class

2

f = Frequency of each class

N = f =∑ Total frequency

iv. Weighted arithmetic mean

= W

WXx

W=∑∑

v. Median

a. For ungrouped data,

If the total of the frequencies is odd, say n

Median = Value of (n 1)

2

+ th item

If total of the frequencies is even, say 2n

Median = Arithmetic Mean of nth and (n + 1)th items

b. For grouped data,

Median = ( ) ( )

mm

N 1 / 2 F 1w L

f

+ − + +

Where,

Lm = Lower limit of the median class

fm = Frequency of the median class

F = Cumulative frequency up to the lower limit of the median class

w = Width of the class interval

N = Total frequency

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vi. Mode (For a grouped data)

Mode = Lmo + mo 1

mo 1 2

f fw

2f f f

−×

− −

Where,

Lmo = Lower limit of the modal class which is the class having the maximum

frequency

f1, f2 = Frequencies of the classes preceding and succeeding the modal class

respectively

fmo = Frequency of modal class

w = Class interval

vii. Empirical mode = 3Median – 2Mean

viii. Geometric mean, G = (X1 x X2 x X3 .... Xn)1

n

ix. Harmonic mean, HM =

1 2 n

N

1 1 1....

x x x+ + +

x. Weighted Harmonic mean, WHM =( )

W

w / x

∑∑

xi. Mean Absolute Deviation = x x

n

−∑

Here, x x− = x x− if x ≥ x

and x x− = x x− if x ≤ x

xii. Quartile Deviation Q.D. = 3 1Q Q

2

−

Where,

Q1 = First quartile = Size of N

th4

observation

Q3 = Third quartile = Size of 3N

th4

observation

N = Number of observations

xiii. Population standard deviation

( )2

x µσ =

N

−∑

Where,

x denotes each observation

µ = Arithmetic mean of population

N = No. of observations

For grouped data,

( )2

f x µσ=

f

−∑∑

Where,

f = Frequency

µ = Arithmetic mean of population

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75

xiv. Sample standard deviation

S = ( )

2x x

n 1

−

−

∑

Where,

x = Sample mean

xv. Combined standard deviation of two groups

2 2 2 21 1 2 2 1 1 2 2

121 2

N N N d N d

N N

σ + σ + +σ =

+

Where,

1µ = Mean of first group

2µ = Mean of second group

1σ = Standard deviation of first group

2σ = Standard deviation of second group

1N = Number of observations in the first group

N2 = Number of observations in the second group

d1 = 1µ – µ

d2 = 2µ – µ

µ = ( )1 1 2 2

1 2

N µ + N µ

N + N

xvi. Standard Deviation of a Discrete Random Variable ( )

1/22n

1 i

i=1

σ= P k k −

∑

Where,

Pi = Probability associated with the occurrence of the ith value

ki = ith possible value

k = Expected rate of return i.e. mean

n = Number of possible outcomes

xvii. Coefficient of variation = Standard Deviation

100Mean

×

5. Probability

i. Marginal or unconditional probability of an event A

P(A) = Number of possibleoutcomes favoring A

Total number of possibleoutcomes

ii. If A and B are mutually exclusive events,

then P(A or B) = P(A) + P(B)

iii. If A and B are not mutually exclusive,

P(A or B) = P(A) + P(B) – P(A and B)

iv. If A and B are independent events,

P(A and B) = P(A) . P(B)

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v. Conditional probability of event A, given that B has occurred, in case of A and B

being independent events is P(A/B) = P(A)

vi. If A and B are dependent then

P(A and B) = P(A) . P(B/A)

or P(B and A) = P(B) . P(A/B)

vii. Bayes’ Theorem:

P(Ai/B) = ( )

( ) ( ) ( ) ( ) ( ) ( )

i i

1 1 2 2 k k

P A P(B / A )

P A P B/ A P A P B/ A ... P A P B/ A+ + +

6. Probability Distribution and Decision Theory

i. Expected Value

E[x] = x∑ P(x),

Where,

x = Random variable

P(x) = Probability of x

ii. Covariance

a. For a population of paired ungrouped data points {x,y}

Covxy = ( )( )x yx µ y µ

N

∑ − −

Where,

xµ = The arithmetic mean of {x}

yµ = The arithmetic mean of {y}

N = The number of observations in each population

For a paired sample {x,y},

Covxy = ( )( )x x y y

n 1

∑ − −

−

Where,

x = The arithmetic mean of sample {x}

y = The arithmetic mean of sample {y}

b. For grouped data of paired population

Covxy = ( )( )x yf x µ y µ

f

∑ − −

∑

Where,

f = The frequency of the corresponding (x,y) values.

Given a probability distribution of paired data {x,y},

Covxy = ( ) ( ) ( )x E x y E y P x, y∑ − −

Where,

P(x,y) = The joint probability of x and y

E(x) = The expected value of x

E(y) = The expected value of y.

iii. E (a1X1+ a2 X2) = a1E(X1) + a2E(X2)

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77

iv. V(a1 X1 + a2 X2) = 21a V(X1) + 2

2a V(X2) + 2a1a2 Cov (X1, X2 )

Where, V denotes variance

v. Binomial distribution,

( )( )n xxn

f (x) p 1 px

− = −

Where,

f(x) = The probability of x successes in n trials

n = The number of trials

n

x

= ( )

n!

x! n x !−

p = The probability of a success on any one trial

(1 – p) = q = The probability of a failure on any one trial

E(x) = np

V(x) = npq

vi. Poisson Distribution

x e

f (x)x!

−λλ ×

=

Where,

f(x) = Probability of x occurrences in an interval

λ = The mean number of occurrences in an interval

e = The base of natural logarithm system

vii. Hypergeometric distribution

r N r

x n xf (x)

N

n

−

− =

for 0 ≤ x ≤ r

Where,

f(x) = Probability of x successes in n trials

n = Number of trials

N = Number of elements in the population

r = Number of elements in the population labeled success

E(x) = nr

N

V(x) = 2

nr(N r)(N n)

N (N 1)

− −

−

viii. Standard Normal Variable,

z = x µ−

σ

Where,

x = Random variable

µ = Mean of the distribution of the random variable

σ = Standard deviation

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ix. In a t-distribution, t = x µ

S/ n

−

Where,

x = The sample mean

µ = The population mean

S = Sample standard deviation

n = The sample size.

x. If MP = Marginal profit

ML = Marginal loss

‘P’ = The probability of generating the additional profit by increasing our

activity level by one unit, then

Expected (MP) = P × MP

Expected (ML) = (1 – P) × ML

P* = ML

ML MP+

P* represents the minimum required probability of selling at least one additional unit

to justify the stocking of that additional unit.

7. Statistical Inferences

i. Standard Error for

a. Sample mean ( x ), xσ = σ

n

b. Sample proportion ( p ), p

pqσ =

n

Where, q = 1 – p

c. Difference of two sample means 1x and 2x i.e.

1 2

2 21 2

X X1 2

σ σσ

n n−

= +

Where,

1x and 2x are the means of two random samples of sizes and drawn from

two populations with standard deviations 1σ and 2σ respectively.

d. Difference of two proportions:

21P P1 2

ˆ ˆ ˆ ˆpq pqσ

n n−

= +

Where,

1p and 2p are the proportions of two random samples of sizes n1 and n2

drawn from two populations and 1 1 2 2

1 2

n p n pp

n n

+=

+and q = 1 – p

Where,

p is the estimate of the overall proportion of success in the combined

populations using combined proportions for both the samples.

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e. For a finite population of size N, when a sample of size n is drawn without

replacement,

σ N n

σx N 1n

−=

−

f. Sample standard deviation,

2

s

σσ

2n=

8. Simple Linear Regression and Correlation i. Karl Pearson’s correlation coefficient,

r = x y

Cov(X,Y)

s s

Or

r = (X X)(Y Y)

2 2(X X) (Y Y)

∑ − −

∑ − ∑ −

ii. Rank correlation coefficient,

n2

ii 1

3

6

R 1n n

D=

= −−

∑

Where, Di = The difference in the ranks of the ith individual

iii. If the regression line, XY a bX= +

b = ( )( )

( )22

n XY X Y

n X X

∑ − ∑ ∑

∑ − ∑

a = Y bX−

iv. Standard error of the estimate for a simple regression equation

Se = ( )

2ˆY Y

n 2

∑ −

− =

2Y a Y b XY

n 2

−Σ Σ − Σ

−

Where,

Y = Values of dependent variable

Y = Estimated values from the estimating equation that correspond to each

Y value

n = Number of data points used to fit the regression line

v. Total Sum of Squares, TSS = ( )2

Y YΣ −

Regression Sum of Squares, RSS = ( )2

Y YΣ −

Error Sum of Squares, ESS = ( )2

ˆY YΣ −

vi. Coefficient of Determination, R2 =

2

2 2

a Y b XY nY

Y nY

Σ + Σ −

Σ −

vii. Sb = e

2 2

S

x nxΣ −

Where, Sb = Estimate of V(b).

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9. Multiple Regression

i. Multiple Regression Equation

Y = A + B1X1+ B2X2 + B3X3 + ….. + BnXk

ii. Standard error of estimate for a multiple regression equation

Se = ( )

2ˆY Y

(n k 1)

Σ −

− −

= 2

1 1 2 2Y a Y b X Y b X Y

n k 1

Σ − Σ − Σ − Σ

− −

Where,

Y = The sample value of the dependent variable

= The corresponding estimate obtained by using the regression equation

n = number of observations

k = number of independent variables

iii. Coefficient of multiple correlation between Y and both X1 and X2 is given by

1 2Y.X XR = 1 – ( )

( )

21 1 2 2

2 2

Y a Y b X Y b X Y

Y ( Y) / n

Σ − Σ − Σ − Σ

Σ − Σ

R1.23 =2 2

12 13 12 23 13

223

r r 2r r r

1 r

+ −

−

iv. In the equation, Y = a + b1X1 + b2X2,

the partial correlation coefficient is given by123

R

Where,

123R =

( )( )

12 13 23

2 213 23

r r .r

1 r 1 r

−

− −

is the partial correlation coefficient between Y and X1,

when X2 is kept constant.

Where,

r12 = Correlation coefficient between Y and X1

r23 = Correlation coefficient between X1 and X2

r13 = Correlation coefficient between Y and X2

R123 will take values between 0 and 1, i.e., 0 ≤ R123 ≤ 1.

10. Time Series Analysis

i. Secular Trend

Using regression analysis, estimating equation is

Y = a + bX (linear trend)

After coding (or translating time)

a = Y b = 2

xY

x

Σ

Σ

Where,

x = ( X X− ) if there are odd number of data points and

x = 2 ( X X− ) if there are even number of data points.

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81

Curvilinear trend, 2Y a bX X= + +

After coding (or translating time) is done

ΣY = an + cΣx2

Σx2Y = aΣx

2 + cΣx

4 and b =

2

xY

x

Σ

Σ

Where,

x = ( X X− ) if there are odd number of data points and

x = 2 ( X X− ) if there are even number of data points

ii. Cyclical Variation

a. Percent of Trend Measure

Cyclical variation component =Y

Y×100

Where,

Y represents actual values and

Y represents estimated values

b. Relative Cyclical Residual Measure

Cyclical Component = ˆY Y

Y

−× 100

11. Index Numbers

i. Unweighted Aggregates Price Index = 1

0

P

P

Σ

Σ× 100

Where,

ΣP1 = Sum of all elements in the composite for current year

ΣP0 = Sum of all elements in the composite for base year

ii. Weighted Aggregates Index

a. Laspeyre’s Price Index

= 1 0

0 0

P Q

P Q

Σ

Σx 100

b. Laspeyre’s Quantity Index = 1 0

0 0

Q P

Q P

Σ

Σx 100

Where,

P1 = Prices in the current year

P0 = Prices in the base year

Q0 = Quantities in the base year

Q1 = Quantities in the current year

c. Paasche’s Price Index

= 1 1

0 1

P Q

P Q

Σ

Σx 100

d. Fisher’s Ideal Price Index = 1 0 1 1

0 0 0 1

P Q P Q

P Q P Q

Σ Σ×

Σ Σ× 100

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e. Fisher’s Ideal Quantity Index = 1 0 1 1

0 0 0 1

Q P Q P

Q P Q P

Σ Σ×

Σ Σ×100

f. Marshall Edgeworth Price Index = 0 1 1

0 1 0

(Q Q ) P

(Q Q ) P

Σ +

Σ +x 100

iii. Value Index Number = 1 1

0 0

P Q

P Q

Σ

Σx 100

iv. Average of Relatives Method

a. Unweighted average of relatives method =

1

0

P100

P

n

Σ ×

b. Unweighted average of relatives quantity index

=

1

0

Q100

Q

n

Σ ×

c. Weighted average of relatives price index

=

1n n

0

n n

P100 (P Q )

P

P Q

Σ ×

Σ

Where,

Pn = Prices in the fixed period

Qn = Quantities in the fixed period

PnQn = Value in the fixed period

v. Chain Index Numbers

Chain Index for a given year =

Average link relative Chain index of

of thegiven year previous year

100

×

Where,

Link relative = Price in a given period

Previous year's price x 100

12. Quality Control

i. X -Charts

a. When Mean and Standard Deviation are known:

Lower limit = x xµ 3σ−

Upper limit = x xµ 3σ+

b. When the mean and standard deviation are not known, then

Lower control limit = 2X A R−

Upper control limit = 2X A R+

Where,

1

X Xk

= Σ = x

n k

Σ

×

A2 =

2

3

d n

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ii. R-Charts

Lower control limit = D3 R

Upper control limit = D4 R

Where,

D3 = 3

2

3d1

d

−

, D4 = 3

2

3d1

d

+

iii. p-Charts

When p is known:

Lower control limit = pp 3− σ

Upper control limit = pp 3+ σ

Where,

p

pqσ

n=

When p is unknown:

jpp

k

Σ=

Where,

jp is the jth sample fraction

k is the number of all the samples considered

In calculating the lower and upper control limits p is used instead of p.

13. Chi-Square Test and Analysis of Variance

i. The chi-square statistic is given by

2

2 0 e

e

(f f )χ

f

−=∑

Where,

f0 = The observed frequency

fe = The expected frequency

ii. Number of Degrees of Freedom in a contingency table

= (Number of rows – 1) × (Number of columns – 1)

Number of degrees of freedom in chi-square test of goodness of fit for ‘k’ data

points = k – 1

iii. ANOVA

a. Between-Column Variance, ( )

2

j j2n x x

ˆk 1

Σ −

σ =−

b. Within column variance,j2 2

jT

n 1ˆ S

n k

− σ =Σ

−

Where,

2σ = The second estimate of the population variance

nj = The size of the jth sample

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84

nT = The total number of elements present in all the samples

k = The number of samples

2

Sj = The sample variance of the sample j

jx = Mean of jth sample

x = Grand mean

c. F ratio =

Population varianceobtained from the varianceamong the

sample means (between column variance)

Population varianceobtained from the variance within

the individualsamples (within column variance)

Degrees of freedom for the numerator = (k – 1)

DOF for Denominator = n

j

k 1

(n 1)=

−∑ = nT – k

iv. Chi-square statistic for a sample variance is given by 2

2

2

(n 1)s−χ =

σ

Number of degrees of freedom = n – 1

v. Inferences about two population variances

F ratio for testing the equality of two population variances is given by

F = 21

22

s

s

Where,

21s = The variance of the first sample

22s = The variance of the second sample

This distribution will have n1 – 1 degrees of freedom in the numerator and n2 – 1

degrees of freedom in the denominator respectively, n1 and n2 represent the number

of elements present in each of the samples.

ICFAI

XI. Security Analysis

1. Bond Valuation

i. The intrinsic value or the present value of a bond

V0 or P0 = I(PVIFAkd, n) + F(PVIFkd, n)

Where,

V0 = Intrinsic value of the bond





kd = Cost of Capital or Required rate of return

ii. Current yield = Coupon Interest

Prevailing Market Price

iii. Yield to maturity is r in the equation, P0 = n

t nt 1

I F

(1 r) (1 r)=

++ +

∑

Where,





iv. Realized yield is r in the equation = P0 (1 + r)n

= Total cash flows received by the investor

v. Nominal Rate = Risk-free rate + Inflation rate

vi. Duration = r,1 r,2 r,n

0

1C.PVIF 2C.PVIF ... n[C F]PVIF

P

+ + + +

Where,

C = Coupon interest payments

r = Promised yield to maturity

n = Number of years to maturity

F = Redemption value

vii. Simplified formula for duration

D = c

d

r

r PVIFA(rd,n) x (1 + rd)+

c

d

r1

r

−

n

Where,

rc = Coupon yield

rd = YTM


viii. When bond is selling at par, (i.e. rc = rd)

Duration (D) = PVIFA(rd, n) × (1 + rd)

Where,

rc = Coupon yield

rd = YTM


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86

ix. Duration of a perpetual bond, D = 1 r

r

+

Where,

r = Current yield

x. Limiting value of duration = 1 YTM

YTM

+

xi. Interest rate elasticity, IE = 0 0P / P

YTM / YTM

∆

∆

Where,

∆ P0 = Change in price for bond in period t

P0 = Price of the bond at the period 0

∆ YTM = Change in YTM for the bond

YTM = Yield to maturity

xii. Approximate method of calculating interest rate elasticity

IE = Dit × YTM

1 YTM+

Where,

Dit = Duration


xiii. Interest rate risk which measures change in price of bond for a change in the YTM

0

0

P

P

∆ = IEit ×

YTM

YTM

∆

Where,

IEit = Interest rate elasticity

∆ P0 = Change in price for bond in period t

P0 = Price of the bond at the period 0

∆ YTM = Change in YTM for the bond


xiv. Modified Duration: Dmod = D

YTM1

f+

Where,

f = Discounting periods per year

D = Macaulay’s duration

YTM = Yield to maturity in decimal form

xv. Percentage price volatility = P

P

∆ × 100 = – Dmod.∆y

Where,

P∆ = Change in the price of the bond

P = Price of the bond

∆y = Change in YTM

Dmod = Modified duration

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87

xvi. Duration of equity based on dividend discount model = 1

k g−

Where,

k = Return required by equity holders

g = Constant growth rate of dividend

xvii. Duration of equity =1

Dividend yield =

Market price

Dividend

2. Equity Stock Valuation Model

i. The intrinsic value or present value equity share

(P0) = ( )

nt n

t nt 1 ee

D P

(1 k )1 k=

+++

∑

Where,

P0 = Current market price of the equity share or intrinsic value of the share

Dt = Expected equity dividend at time t

Pn = Expected price of the equity share at time n


ii. The value of equity share when there is constant growth

(P0) = 0

e

D (1 g)

k g

+

−

Where,

P0 = Intrinsic value of the share


g = Expected constant growth rate in dividends


iii. The value of equity share using H Model

(P0) = [ ]0 n a n

n

D (1 g ) H (g g )

r g

+ + −

−

Where,

P0 = Intrinsic value of the share


r = Required rate of return

gn = Normal long run growth rate

ga = Current growth rate

H = One half of the period during which ga will level off to gn

3. Technical Analysis

i. Relative Strength, RS = Averageof ' x 'days up-closings

Averageof 'x 'days down-closings

ii. Relative strength index = 100 – 100

1 RS+

iii. Odd-lot index = Odd-lot sales

Odd-lot purchases

ICFAI

Formulae and Tables

88

iv. Odd-lot short sales ratio = Odd-lot short sales

Totalodd-lot sales

v. Stochastics (% K) = C L

H L

−

−x 100

Where,

C = Latest closing price

L = Low price during the last N periods

H = High price during the last N periods

N = Number of periods

% D = Derived by smoothening % K using the simple moving average technique.

4. Warrants and Convertibles

i. Percentage of downside risk =

Market price of convertible security Price of an equivalent non-convertible

security×100

Price of an equivalent non-convertible security

−

ii. Conversion premium = Market price Conversion value

100Conversion value

−×

iii. Conversion parity price = Bond price

Number of shareson conversion per warrant

iv. Break even period = Conversion premium

Interest income Dividends−

v. Payback period =

% premium

1 % premium

Dividend yieldCurrent yield

1 % premium

+

−+

5. Real Assets and Mutual Funds

i. MV0 =

nt n

t nt 1

NOI MV

(1 r) (1 r)=

++ +

∑

Where,

MV0 = The current market price of the property

MVn = The expected sales price of the property

r = The required rate of return

NOIt = Net operating income at time t

ii. If operating income grows at the rate ‘g’ annually,

MV0 = NOI

r g−

Where,

NOI = Net operating income

g = Growth rate

r = Required return

iii. Net Asset Value (NAV) = Assets Liabilities

No.of units outstanding

−

ICFAI

XII. Strategic Financial Management

1. Capital Structure

i. Relation between EBIT and EPS

EPS = (EBIT I)(1 t)

n

− −

Where,

EBIT = Earning Before Interest and Tax

EPS = Earning per share

I = Interest payment

t = Tax rate

n = Number of shares

ii. EBIT – EPS Indifference Point = 1 2

1 2

(EBIT I )(1 t) (EBIT I )(1 t)

n n

− − − −=

Where,

EPS = Earning Per Share

I1 & I2 = Interest payment under alternative one and interest payment under

alternative two respectively

iii. Relation between ROI and ROE

ROE = {ROI + (ROI – kd) D/E} (I – t)

Where,

ROE = The Return on Equity

ROI = The Return on Investment

kd = The cost of debt (pre-tax)

D = The debt component in the total capital

E = The equity component in the total capital

t = The tax rate.

2. Decision Support Models

i. Extended Probabilistic Analysis

C1 = 0C ns vns nf ni T(ns vns nf ni nf )′+ − − − − − − − −% % % % % % % % % % % % %

Where,

C1 = Ending cash balance

C0 = Beginning cash balance

n% = Duration of the recession in months

s% = Monthly sales during the recession

ns% % = Total sales during the recession

v = Proportion of variable cash expenses to sales

vns% % = Total variable cash expenses during the recession

ICFAI

Formulae and Tables

90

f = Monthly fixed cash expenses, other than debt servicing burden, during

the recession

nf% = Total fixed cash expenses, other than debt servicing burden during the

recession

i = Monthly interest payment associated with the contemplated level of debt

during the recession

ni% = Total interest payment associated with the contemplated level of debt

during the recession

f ′ = Monthly non-cash fixed expenses

nf ′ = Total non-cash fixed expenses during the recession

T = Corporate income tax rate.

3. Working Capital Management

i. Discriminant Analysis

Zi = aXi + bYi

Where,

Zi = The Z-score for the ith account

Xi = The value of the first independent variable for the ith account

Yi = The value of the second independent variable for the ith account a and

b are the parameter values

a =

( )

2y x xy y

22 2x y xy

.d .d

.

σ − σ

σ σ − σ

b =

( )

2x y xy x

22 2x y xy

.d .d

.

σ − σ

σ σ − σ

Where,

2xσ = Variance of X (across groups 1 and 2)

σxy = Covariance of X and Y (across groups 1 and 2)

2yσ = Variance of Y (across groups 1 and 2)

dx = Difference between the mean values of X for groups 1 and 2

dy = Difference between the mean values of Y for groups 1 and 2

ii. Cash Management Models

a. Baumol Model, TC = I (C/2) + b (T/C)

Where,

TC = Total costs (total conversion costs + total holding costs)

I = Interest rate on marketable securities per planning period

C = Amount of securities liquidated per batch

T = Estimated cash requirement over the planning period

The point where total costs are minimum:

C = 2bT

I


ICFAI

Strategic Financial Management

91

b. Miller and Orr Model

RP = 2

33bσ

4I+ LL and,

UL = 3 RP – 2 LL

Where,

LL = Lower control limit

RP = Return point

UL = Upper control limit


I = Interest rate per day on marketable securities

σ2 = Variance of daily changes in the expected cash balance.

4. Firms in Financial Distress

i. Altman’s Z-Score Model (to identify the financial distress of the firm)

Z = 1.2X1 + 1.4X2 + 3.3X3 + 0.6X4 + 1.0X5

Where,

Z = Discriminant score

X1 = Working capital/Total assets

X2 = Retained earnings/Total assets

X3 = EBIT/Total assets

X4 = Market value of equity/Book value of debt

X5 = Sales/Total assets.

5. Valuation of Firms

i. a. Free cashflow of a firm = Free cashflow from operations + Non-

operating cashflows

b. Free cash flows from operations = Gross cash flows of the firm – Gross

investments

c. Gross cashflows of the firm = EBIT (1– T) + Depreciation + Non-cash

charges

d. Gross Investment = Increase in Net Working Capital + Capital

Expenditure incurred + Increase in Other

Assets.

6. Mergers and Acquisitions

i. Net Acquisition Value

NAV = PVab – (PVa + PVb) – P – E

Where,

NAV = The Net Acquisition Value

PVab = The present value of the merged entity

PVa = The present value of firm A

PVb = The present value of firm B

P = The premium paid by Firm A to acquire Firm B

E = The expenses involved in the merger

ICFAI

Formulae and Tables

92

ii. Conn & Nielson Model

a. Maximum Exchange Ratio acceptable to the Acquiring company

ER = 1 1 2 12

2 1 2

S (E E ) PE

S P S

− ++

Where,

ER = Exchange Ratio

E1 & E2 = Earnings per Share of acquiring and target companies

respectively

P1 = Market price per share of acquiring company

PE12 = Price to Earnings Multiple of merged entity

S1 & S2 = Number of Shares outstanding in acquiring and target

companies respectively

b. Minimum Exchange Ratio Acceptable to the Target Company

ER = ( )( )

2 1

12 1 2 2 2

P S

PE E E P S+ −

Where,

ER = Exchange Ratio

P2 = Market price per share of target company

E1 & E2 = Earnings per share of acquiring and target companies

respectively

PE12 = Price to Earnings Multiple of merged entity

S1 & S2 = Number of shares outstanding in acquiring and target companies

respectively.

ICFAI

TABLES

ICFAI

INTEREST RATE TABLES

Table A.1: Future Value Interest Factor

FV = PV(1 + k)n

n/i 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.0%

1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 1.0800 1.0900 1.1000

2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 1.1664 1.1881 1.2100

3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 1.2597 1.2950 1.3310

4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 1.3605 1.4116 1.4641

5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 1.4693 1.5386 1.6105

6 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 1.5007 1.5869 1.6771 1.7716

7 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 1.7138 1.8280 1.9487

8 1.0829 1.1717 1.2668 1.3686 1.4775 1.5938 1.7182 1.8509 1.9926 2.1436

9 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 1.9990 2.1719 2.3579

10 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672 2.1589 2.3674 2.5937

11 1.1157 1.2434 1.3842 1.5395 1.7103 1.8983 2.1049 2.3316 2.5804 2.8531

12 1.1268 1.2682 1.4258 1.6010 1.7959 2.0122 2.2522 2.5182 2.8127 3.1384

13 1.1381 1.2936 1.4685 1.6651 1.8856 2.1329 2.4098 2.7196 3.0658 3.4523

14 1.1495 1.3195 1.5126 1.7317 1.9799 2.2609 2.5785 2.9372 3.3417 3.7975

15 1.1610 1.3459 1.5580 1.8009 2.0789 2.3966 2.7590 3.1722 3.6425 4.1772

16 1.1726 1.3728 1.6047 1.8730 2.1829 2.5404 2.9522 3.4259 3.9703 4.5950

17 1.1843 1.4002 1.6528 1.9479 2.2920 2.6928 3.1588 3.7000 4.3276 5.0545

18 1.1961 1.4282 1.7024 2.0258 2.4066 2.8543 3.3799 3.9960 4.7171 5.5599

19 1.2081 1.4568 1.7535 2.1068 2.5270 3.0256 3.6165 4.3157 5.1417 6.1159

20 1.2202 1.4859 1.8061 2.1911 2.6533 3.2071 3.8697 4.6610 5.6044 6.7275

21 1.2324 1.5157 1.8603 2.2788 2.7860 3.3996 4.1406 5.0338 6.1088 7.4002

22 1.2447 1.5460 1.9161 2.3699 2.9253 3.6035 4.4304 5.4365 6.6586 8.1403

23 1.2572 1.5769 1.9736 2.4647 3.0715 3.8197 4.7405 5.8715 7.2579 8.9543

24 1.2697 1.6084 2.0328 2.5633 3.2251 4.0489 5.0724 6.3412 7.9111 9.8497

25 1.2824 1.6406 2.0938 2.6658 3.3864 4.2919 5.4274 6.8485 8.6231 10.8347

26 1.2953 1.6734 2.1566 2.7725 3.5557 4.5494 5.8074 7.3964 9.3992 11.9182

27 1.3082 1.7069 2.2213 2.8834 3.7335 4.8223 6.2139 7.9881 10.2451 13.1100

28 1.3213 1.7410 2.2879 2.9987 3.9201 5.1117 6.6488 8.6271 11.1671 14.4210

29 1.3345 1.7758 2.3566 3.1187 4.1161 5.4184 7.1143 9.3173 12.1722 15.8631

30 1.3478 1.8114 2.4273 3.2434 4.3219 5.7435 7.6123 10.0627 13.2677 17.4494

40 1.4889 2.2080 3.2620 4.8010 7.0400 10.2857 14.9745 21.7245 31.4094 45.2593

50 1.6446 2.6916 4.3839 7.1067 11.4674 18.4202 29.4570 46.9016 74.3575 117.3909

60 1.8167 3.2810 5.8916 10.5196 18.6792 32.9877 57.9464 101.2571 176.0313 304.4816

ICFAI

Formulae and Tables

96

n/i 12.0% 14.0% 15.0% 16.0% 18.0% 20.0% 24.0% 28.0% 32.0% 36.0%

1 1.1200 1.1400 1.1500 1.1600 1.1800 1.2000 1.2400 1.2800 1.3200 1.3600

2 1.2544 1.2996 1.3225 1.3456 1.3924 1.4400 1.5376 1.6384 1.7424 1.8496

3 1.4049 1.4815 1.5209 1.5609 1.6430 1.7280 1.9066 2.0972 2.3000 2.5155

4 1.5735 1.6890 1.7490 1.8106 1.9388 2.0736 2.3642 2.6844 3.0360 3.4210

5 1.7623 1.9254 2.0114 2.1003 2.2878 2.4883 2.9316 3.4360 4.0075 4.6526

6 1.9738 2.1950 2.3131 2.4364 2.6996 2.9860 3.6352 4.3980 5.2899 6.3275

7 2.2107 2.5023 2.6600 2.8262 3.1855 3.5832 4.5077 5.6295 6.9826 8.6054

8 2.4760 2.8526 3.0590 3.2784 3.7589 4.2998 5.5895 7.2058 9.2170 11.7034

9 2.7731 3.2519 3.5179 3.8030 4.4355 5.1598 6.9310 9.2234 12.1665 15.9166

10 3.1058 3.7072 4.0456 4.4114 5.2338 6.1917 8.5944 11.8059 16.0598 21.6466

11 3.4785 4.2262 4.6524 5.1173 6.1759 7.4301 10.6571 15.1116 21.1989 29.4393

12 3.8960 4.8179 5.3503 5.9360 7.2876 8.9161 13.2148 19.3428 27.9825 40.0375

13 4.3635 5.4924 6.1528 6.8858 8.5994 10.6993 16.3863 24.7588 36.9370 54.4510

14 4.8871 6.2613 7.0757 7.9875 10.1472 12.8392 20.3191 31.6913 48.7568 74.0534

15 5.4736 7.1379 8.1371 9.2655 11.9737 15.4070 25.1956 40.5648 64.3590 100.7126

16 6.1304 8.1372 9.3576 10.7480 14.1290 18.4884 31.2426 51.9230 84.9538 136.9691

17 6.8660 9.2765 10.7613 12.4677 16.6722 22.1861 38.7408 66.4614 112.1390 186.2779

18 7.6900 10.5752 12.3755 14.4625 19.6733 26.6233 48.0386 85.0706 148.0235 253.3380

19 8.6128 12.0557 14.2318 16.7765 23.2144 31.9480 59.5679 108.8904 195.3911 344.5397

20 9.6463 13.7435 16.3665 19.4608 27.3930 38.3376 73.8641 139.3797 257.9162 468.5740

21 10.8038 15.6676 18.8215 22.5745 32.3238 46.0051 91.5915 178.4060 340.4494 637.2606

22 12.1003 17.8610 21.6447 26.1864 38.1421 55.2061 113.5735 228.3596 449.3932 866.6744

23 13.5523 20.3616 24.8915 30.3762 45.0076 66.2474 140.8312 292.3003 593.1990 1178.6772

24 15.1786 23.2122 28.6252 35.2364 53.1090 79.4968 174.6306 374.1444 783.0227 1603.0010

25 17.0001 26.4619 32.9190 40.8742 62.6686 95.3962 216.5420 478.9049 1033.5900 2180.0814

26 19.0401 30.1666 37.8568 47.4141 73.9490 114.4755 268.5121 612.9982 1364.3387 2964.9107

27 21.3249 34.3899 43.5353 55.0004 87.2598 137.3706 332.9550 784.6377 1800.9271 4032.2786

28 23.8839 39.2045 50.0656 63.8004 102.9666 164.8447 412.8642 1004.3363 2377.2238 5483.8988

29 26.7499 44.6931 57.5755 74.0085 121.5005 197.8136 511.9516 1285.5504 3137.9354 7458.1024

30 29.9599 50.9502 66.2118 85.8499 143.3706 237.3763 634.8199 1645.5046 4142.0748 10143.0193

40 93.0510 188.8835 267.8635 378.7212 750.3783 1469.7716 5455.9126 19426.6889 66520.7670 219561.5736

50 289.0022 700.2330 1083.6574 1670.7038 3927.3569 9100.4382 46890.4346 229349.8616 1068308.1960 4752754.9027

60 897.5969 2595.9187 4383.9987 7370.2014 20555.1400 56347.5144 402996.3473 2707685.2482 17156783.5543 102880840.1651

ICFAI

Tables

97

Table A.2 : Future Value Interest Factor for an Annuity

FVIFA(k, n) = ( )

n1 k 1

k

+ −

n/i 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.0%

1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2 2.0100 2.0200 2.0300 2.0400 2.0500 2.0600 2.0700 2.0800 2.0900 2.1000

3 3.0301 3.0604 3.0909 3.1216 3.1525 3.1836 3.2149 3.2464 3.2781 3.3100

4 4.0604 4.1216 4.1836 4.2465 4.3101 4.3746 4.4399 4.5061 4.5731 4.6410

5 5.1010 5.2040 5.3091 5.4163 5.5256 5.6371 5.7507 5.8666 5.9847 6.1051

6 6.1520 6.3081 6.4684 6.6330 6.8019 6.9753 7.1533 7.3359 7.5233 7.7156

7 7.2135 7.4343 7.6625 7.8983 8.1420 8.3938 8.6540 8.9228 9.2004 9.4872

8 8.2857 8.5830 8.8923 9.2142 9.5491 9.8975 10.2598 10.6366 11.0285 11.4359

9 9.3685 9.7546 10.1591 10.5828 11.0266 11.4913 11.9780 12.4876 13.0210 13.5795

10 10.4622 10.9497 11.4639 12.0061 12.5779 13.1808 13.8164 14.4866 15.1929 15.9374

11 11.5668 12.1687 12.8078 13.4864 14.2068 14.9716 15.7836 16.6455 17.5603 18.5312

12 12.6825 13.4121 14.1920 15.0258 15.9171 16.8699 17.8885 18.9771 20.1407 21.3843

13 13.8093 14.6803 15.6178 16.6268 17.7130 18.8821 20.1406 21.4953 22.9534 24.5227

14 14.9474 15.9739 17.0863 18.2919 19.5986 21.0151 22.5505 24.2149 26.0192 27.9750

15 16.0969 17.2934 18.5989 20.0236 21.5786 23.2760 25.1290 27.1521 29.3609 31.7725

16 17.2579 18.6393 20.1569 21.8245 23.6575 25.6725 27.8881 30.3243 33.0034 35.9497

17 18.4304 20.0121 21.7616 23.6975 25.8404 28.2129 30.8402 33.7502 36.9737 40.5447

18 19.6147 21.4123 23.4144 25.6454 28.1324 30.9057 33.9990 37.4502 41.3013 45.5992

19 20.8109 22.8406 25.1169 27.6712 30.5390 33.7600 37.3790 41.4463 46.0185 51.1591

20 22.0190 24.2974 26.8704 29.7781 33.0660 36.7856 40.9955 45.7620 51.1601 57.2750

21 23.2392 25.7833 28.6765 31.9692 35.7193 39.9927 44.8652 50.4229 56.7645 64.0025

22 24.4716 27.2990 30.5368 34.2480 38.5052 43.3923 49.0057 55.4568 62.8733 71.4027

23 25.7163 28.8450 32.4529 36.6179 41.4305 46.9958 53.4361 60.8933 69.5319 79.5430

24 26.9735 30.4219 34.4265 39.0826 44.5020 50.8156 58.1767 66.7648 76.7898 88.4973

25 28.2432 32.0303 36.4593 41.6459 47.7271 54.8645 63.2490 73.1059 84.7009 98.3471

26 29.5256 33.6709 38.5530 44.3117 51.1135 59.1564 68.6765 79.9544 93.3240 109.1818

27 30.8209 35.3443 40.7096 47.0842 54.6691 63.7058 74.4838 87.3508 102.7231 121.0999

28 32.1291 37.0512 42.9309 49.9676 58.4026 68.5281 80.6977 95.3388 112.9682 134.2099

29 33.4504 38.7922 45.2189 52.9663 62.3227 73.6398 87.3465 103.9659 124.1354 148.6309

30 34.7849 40.5681 47.5754 56.0849 66.4388 79.0582 94.4608 113.2832 136.3075 164.4940

40 48.8864 60.4020 75.4013 95.0255 120.7998 154.7620 199.6351 259.0565 337.8824 442.5926

50 64.4632 84.5794 112.7969 152.6671 209.3480 290.3359 406.5289 573.7702 815.0836 1163.9085

60 81.6697 114.0515 163.0534 237.9907 353.5837 533.1282 813.5204 1253.2133 1944.7921 3034.8164

ICFAI

Formulae and Tables

98

n/i 12.0% 14.0% 15.0% 16.0% 18.0% 20.0% 24.0% 28.0% 32.0% 36.0%

1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2 2.1200 2.1400 2.1500 2.1600 2.1800 2.2000 2.2400 2.2800 2.3200 2.3600

3 3.3744 3.4396 3.4725 3.5056 3.5724 3.6400 3.7776 3.9184 4.0624 4.2096

4 4.7793 4.9211 4.9934 5.0665 5.2154 5.3680 5.6842 6.0156 6.3624 6.7251

5 6.3528 6.6101 6.7424 6.8771 7.1542 7.4416 8.0484 8.6999 9.3983 10.1461

6 8.1152 8.5355 8.7537 8.9775 9.4420 9.9299 10.9801 12.1359 13.4058 14.7987

7 10.0890 10.7305 11.0668 11.4139 12.1415 12.9159 14.6153 16.5339 18.6956 21.1262

8 12.2997 13.2328 13.7268 14.2401 15.3270 16.4991 19.1229 22.1634 25.6782 29.7316

9 14.7757 16.0853 16.7858 17.5185 19.0859 20.7989 24.7125 29.3692 34.8953 41.4350

10 17.5487 19.3373 20.3037 21.3215 23.5213 25.9587 31.6434 38.5926 47.0618 57.3516

11 20.6546 23.0445 24.3493 25.7329 28.7551 32.1504 40.2379 50.3985 63.1215 78.9982

12 24.1331 27.2707 29.0017 30.8502 34.9311 39.5805 50.8950 65.5100 84.3204 108.4375

13 28.0291 32.0887 34.3519 36.7862 42.2187 48.4966 64.1097 84.8529 112.3030 148.4750

14 32.3926 37.5811 40.5047 43.6720 50.8180 59.1959 80.4961 109.6117 149.2399 202.9260

15 37.2797 43.8424 47.5804 51.6595 60.9653 72.0351 100.8151 141.3029 197.9967 276.9793

16 42.7533 50.9804 55.7175 60.9250 72.9390 87.4421 126.0108 181.8677 262.3557 377.6919

17 48.8837 59.1176 65.0751 71.6730 87.0680 105.9306 157.2534 233.7907 347.3095 514.6610

18 55.7497 68.3941 75.8364 84.1407 103.7403 128.1167 195.9942 300.2521 459.4485 700.9389

19 63.4397 78.9692 88.2118 98.6032 123.4135 154.7400 244.0328 385.3227 607.4721 954.2769

20 72.0524 91.0249 102.4436 115.3797 146.6280 186.6880 303.6006 494.2131 802.8631 1298.8166

21 81.6987 104.7684 118.8101 134.8405 174.0210 225.0256 377.4648 633.5927 1060.7793 1767.3906

22 92.5026 120.4360 137.6316 157.4150 206.3448 271.0307 469.0563 811.9987 1401.2287 2404.6512

23 104.6029 138.2970 159.2764 183.6014 244.4868 326.2369 582.6298 1040.3583 1850.6219 3271.3256

24 118.1552 158.6586 184.1678 213.9776 289.4945 392.4842 723.4610 1332.6586 2443.8209 4450.0029

25 133.3339 181.8708 212.7930 249.2140 342.6035 471.9811 898.0916 1706.8031 3226.8436 6053.0039

26 150.3339 208.3327 245.7120 290.0883 405.2721 567.3773 1114.6336 2185.7079 4260.4336 8233.0853

27 169.3740 238.4993 283.5688 337.5023 479.2211 681.8528 1383.1457 2798.7061 5624.7723 11197.9960

28 190.6989 272.8892 327.1041 392.5028 566.4809 819.2233 1716.1007 3583.3438 7425.6994 15230.2745

29 214.5828 312.0937 377.1697 456.3032 669.4475 984.0680 2128.9648 4587.6801 9802.9233 20714.1734

30 241.3327 356.7868 434.7451 530.3117 790.9480 1181.8816 2640.9164 5873.2306 12940.8587 28172.2758

40 767.0914 1342.0251 1779.0903 2360.7572 4163.2130 7343.8578 22728.8026 69377.4604 207874.2719 609890.4824

50 2400.0182 4994.5213 7217.7163 10435.6488 21813.0937 45497.1908 195372.6442 819103.0771 3338459.9875 13202094.1741

60 7471.6411 18535.1333 29219.9916 46057.5085 114189.6665 281732.5718 1679147.2802 9670300.8863 53614945.4823285780108.7920

ICFAI

Tables

99

Table A.3 : Present Value Interest Factor

PV = ( )

n

1

1 k+

n/i 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.0%

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091

2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264

3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513

4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830

5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209

6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 0.6302 0.5963 0.5645

7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 0.5835 0.5470 0.5132

8 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 0.5403 0.5019 0.4665

9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 0.5002 0.4604 0.4241

10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855

11 0.8963 0.8043 0.7224 0.6496 0.5847 0.5268 0.4751 0.4289 0.3875 0.3505

12 0.8874 0.7885 0.7014 0.6246 0.5568 0.4970 0.4440 0.3971 0.3555 0.3186

13 0.8787 0.7730 0.6810 0.6006 0.5303 0.4688 0.4150 0.3677 0.3262 0.2897

14 0.8700 0.7579 0.6611 0.5775 0.5051 0.4423 0.3878 0.3405 0.2992 0.2633

15 0.8613 0.7430 0.6419 0.5553 0.4810 0.4173 0.3624 0.3152 0.2745 0.2394

16 0.8528 0.7284 0.6232 0.5339 0.4581 0.3936 0.3387 0.2919 0.2519 0.2176

17 0.8444 0.7142 0.6050 0.5134 0.4363 0.3714 0.3166 0.2703 0.2311 0.1978

18 0.8360 0.7002 0.5874 0.4936 0.4155 0.3503 0.2959 0.2502 0.2120 0.1799

19 0.8277 0.6864 0.5703 0.4746 0.3957 0.3305 0.2765 0.2317 0.1945 0.1635

20 0.8195 0.6730 0.5537 0.4564 0.3769 0.3118 0.2584 0.2145 0.1784 0.1486

21 0.8114 0.6598 0.5375 0.4388 0.3589 0.2942 0.2415 0.1987 0.1637 0.1351

22 0.8034 0.6468 0.5219 0.4220 0.3418 0.2775 0.2257 0.1839 0.1502 0.1228

23 0.7954 0.6342 0.5067 0.4057 0.3256 0.2618 0.2109 0.1703 0.1378 0.1117

24 0.7876 0.6217 0.4919 0.3901 0.3101 0.2470 0.1971 0.1577 0.1264 0.1015

25 0.7798 0.6095 0.4776 0.3751 0.2953 0.2330 0.1842 0.1460 0.1160 0.0923

26 0.7720 0.5976 0.4637 0.3607 0.2812 0.2198 0.1722 0.1352 0.1064 0.0839

27 0.7644 0.5859 0.4502 0.3468 0.2678 0.2074 0.1609 0.1252 0.0976 0.0763

28 0.7568 0.5744 0.4371 0.3335 0.2551 0.1956 0.1504 0.1159 0.0895 0.0693

29 0.7493 0.5631 0.4243 0.3207 0.2429 0.1846 0.1406 0.1073 0.0822 0.0630

30 0.7419 0.5521 0.4120 0.3083 0.2314 0.1741 0.1314 0.0994 0.0754 0.0573

40 0.6717 0.4529 0.3066 0.2083 0.1420 0.0972 0.0668 0.0460 0.0318 0.0221

50 0.6080 0.3715 0.2281 0.1407 0.0872 0.0543 0.0339 0.0213 0.0134 0.0085

60 0.5504 0.3048 0.1697 0.0951 0.0535 0.0303 0.0173 0.0099 0.0057 0.0033

ICFAI

Formulae and Tables

100

n/i 12.0% 14.0% 15.0% 16.0% 18.0% 20.0% 24.0% 28.0% 32.0% 36.0%

1 0.8929 0.8772 0.8696 0.8621 0.8475 0.8333 0.8065 0.7813 0.7576 0.7353

2 0.7972 0.7695 0.7561 0.7432 0.7182 0.6944 0.6504 0.6104 0.5739 0.5407

3 0.7118 0.6750 0.6575 0.6407 0.6086 0.5787 0.5245 0.4768 0.4348 0.3975

4 0.6355 0.5921 0.5718 0.5523 0.5158 0.4823 0.4230 0.3725 0.3294 0.2923

5 0.5674 0.5194 0.4972 0.4761 0.4371 0.4019 0.3411 0.2910 0.2495 0.2149

6 0.5066 0.4556 0.4323 0.4104 0.3704 0.3349 0.2751 0.2274 0.1890 0.1580

7 0.4523 0.3996 0.3759 0.3538 0.3139 0.2791 0.2218 0.1776 0.1432 0.1162

8 0.4039 0.3506 0.3269 0.3050 0.2660 0.2326 0.1789 0.1388 0.1085 0.0854

9 0.3606 0.3075 0.2843 0.2630 0.2255 0.1938 0.1443 0.1084 0.0822 0.0628

10 0.3220 0.2697 0.2472 0.2267 0.1911 0.1615 0.1164 0.0847 0.0623 0.0462

11 0.2875 0.2366 0.2149 0.1954 0.1619 0.1346 0.0938 0.0662 0.0472 0.0340

12 0.2567 0.2076 0.1869 0.1685 0.1372 0.1122 0.0757 0.0517 0.0357 0.0250

13 0.2292 0.1821 0.1625 0.1452 0.1163 0.0935 0.0610 0.0404 0.0271 0.0184

14 0.2046 0.1597 0.1413 0.1252 0.0985 0.0779 0.0492 0.0316 0.0205 0.0135

15 0.1827 0.1401 0.1229 0.1079 0.0835 0.0649 0.0397 0.0247 0.0155 0.0099

16 0.1631 0.1229 0.1069 0.0930 0.0708 0.0541 0.0320 0.0193 0.0118 0.0073

17 0.1456 0.1078 0.0929 0.0802 0.0600 0.0451 0.0258 0.0150 0.0089 0.0054

18 0.1300 0.0946 0.0808 0.0691 0.0508 0.0376 0.0208 0.0118 0.0068 0.0039

19 0.1161 0.0829 0.0703 0.0596 0.0431 0.0313 0.0168 0.0092 0.0051 0.0029

20 0.1037 0.0728 0.0611 0.0514 0.0365 0.0261 0.0135 0.0072 0.0039 0.0021

21 0.0926 0.0638 0.0531 0.0443 0.0309 0.0217 0.0109 0.0056 0.0029 0.0016

22 0.0826 0.0560 0.0462 0.0382 0.0262 0.0181 0.0088 0.0044 0.0022 0.0012

23 0.0738 0.0491 0.0402 0.0329 0.0222 0.0151 0.0071 0.0034 0.0017 0.0008

24 0.0659 0.0431 0.0349 0.0284 0.0188 0.0126 0.0057 0.0027 0.0013 0.0006

25 0.0588 0.0378 0.0304 0.0245 0.0160 0.0105 0.0046 0.0021 0.0010 0.0005

26 0.0525 0.0331 0.0264 0.0211 0.0135 0.0087 0.0037 0.0016 0.0007 0.0003

27 0.0469 0.0291 0.0230 0.0182 0.0115 0.0073 0.0030 0.0013 0.0006 0.0002

28 0.0419 0.0255 0.0200 0.0157 0.0097 0.0061 0.0024 0.0010 0.0004 0.0002

29 0.0374 0.0224 0.0174 0.0135 0.0082 0.0051 0.0020 0.0008 0.0003 0.0001

30 0.0334 0.0196 0.0151 0.0116 0.0070 0.0042 0.0016 0.0006 0.0002 0.0001

40 0.0107 0.0053 0.0037 0.0026 0.0013 0.0007 0.0002 0.0001 0.0000 0.0000

50 0.0035 0.0014 0.0009 0.0006 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000

60 0.0011 0.0004 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

ICFAI

Tables

101

Table A.4 : Present Value Interest Factor for an Annuity

PVIFA(k, n) = ( )

n

n

1 k 1

k(1 k)

+ −

+

n/i 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.0%

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091

2 1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355

3 2.9410 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869

4 3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3872 3.3121 3.2397 3.1699

5 4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908

6 5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553

7 6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684

8 7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349

9 8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590

10 9.4713 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446

11 10.3676 9.7868 9.2526 8.7605 8.3064 7.8869 7.4987 7.1390 6.8052 6.4951

12 11.2551 10.5753 9.9540 9.3851 8.8633 8.3838 7.9427 7.5361 7.1607 6.8137

13 12.1337 11.3484 10.6350 9.9856 9.3936 8.8527 8.3577 7.9038 7.4869 7.1034

14 13.0037 12.1062 11.2961 10.5631 9.8986 9.2950 8.7455 8.2442 7.7862 7.3667

15 13.8651 12.8493 11.9379 11.1184 10.3797 9.7122 9.1079 8.5595 8.0607 7.6061

16 14.7179 13.5777 12.5611 11.6523 10.8378 10.1059 9.4466 8.8514 8.3126 7.8237

17 15.5623 14.2919 13.1661 12.1657 11.2741 10.4773 9.7632 9.1216 8.5436 8.0216

18 16.3983 14.9920 13.7535 12.6593 11.6896 10.8276 10.0591 9.3719 8.7556 8.2014

19 17.2260 15.6785 14.3238 13.1339 12.0853 11.1581 10.3356 9.6036 8.9501 8.3649

20 18.0456 16.3514 14.8775 13.5903 12.4622 11.4699 10.5940 9.8181 9.1285 8.5136

21 18.8570 17.0112 15.4150 14.0292 12.8212 11.7641 10.8355 10.0168 9.2922 8.6487

22 19.6604 17.6580 15.9369 14.4511 13.1630 12.0416 11.0612 10.2007 9.4424 8.7715

23 20.4558 18.2922 16.4436 14.8568 13.4886 12.3034 11.2722 10.3711 9.5802 8.8832

24 21.2434 18.9139 16.9355 15.2470 13.7986 12.5504 11.4693 10.5288 9.7066 8.9847

25 22.0232 19.5235 17.4131 15.6221 14.0939 12.7834 11.6536 10.6748 9.8226 9.0770

26 22.7952 20.1210 17.8768 15.9828 14.3752 13.0032 11.8258 10.8100 9.9290 9.1609

27 23.5596 20.7069 18.3270 16.3296 14.6430 13.2105 11.9867 10.9352 10.0266 9.2372

28 24.3164 21.2813 18.7641 16.6631 14.8981 13.4062 12.1371 11.0511 10.1161 9.3066

29 25.0658 21.8444 19.1885 16.9837 15.1411 13.5907 12.2777 11.1584 10.1983 9.3696

30 25.8077 22.3965 19.6004 17.2920 15.3725 13.7648 12.4090 11.2578 10.2737 9.4269

40 32.8347 27.3555 23.1148 19.7928 17.1591 15.0463 13.3317 11.9246 10.7574 9.7791

50 39.1961 31.4236 25.7298 21.4822 18.2559 15.7619 13.8007 12.2335 10.9617 9.9148

60 44.9550 34.7609 27.6756 22.6235 18.9293 16.1614 14.0392 12.3766 11.0480 9.9672

ICFAI

Formulae and Tables

102

n/i 12.0% 14.0% 15.0% 16.0% 18.0% 20.0% 24.0% 28.0% 32.0% 36.0%

1 0.8929 0.8772 0.8696 0.8621 0.8475 0.8333 0.8065 0.7813 0.7576 0.7353

2 1.6901 1.6467 1.6257 1.6052 1.5656 1.5278 1.4568 1.3916 1.3315 1.2760

3 2.4018 2.3216 2.2832 2.2459 2.1743 2.1065 1.9813 1.8684 1.7663 1.6735

4 3.0373 2.9137 2.8550 2.7982 2.6901 2.5887 2.4043 2.2410 2.0957 1.9658

5 3.6048 3.4331 3.3522 3.2743 3.1272 2.9906 2.7454 2.5320 2.3452 2.1807

6 4.1114 3.8887 3.7845 3.6847 3.4976 3.3255 3.0205 2.7594 2.5342 2.3388

7 4.5638 4.2883 4.1604 4.0386 3.8115 3.6046 3.2423 2.9370 2.6775 2.4550

8 4.9676 4.6389 4.4873 4.3436 4.0776 3.8372 3.4212 3.0758 2.7860 2.5404

9 5.3282 4.9464 4.7716 4.6065 4.3030 4.0310 3.5655 3.1842 2.8681 2.6033

10 5.6502 5.2161 5.0188 4.8332 4.4941 4.1925 3.6819 3.2689 2.9304 2.6495

11 5.9377 5.4527 5.2337 5.0286 4.6560 4.3271 3.7757 3.3351 2.9776 2.6834

12 6.1944 5.6603 5.4206 5.1971 4.7932 4.4392 3.8514 3.3868 3.0133 2.7084

13 6.4235 5.8424 5.5831 5.3423 4.9095 4.5327 3.9124 3.4272 3.0404 2.7268

14 6.6282 6.0021 5.7245 5.4675 5.0081 4.6106 3.9616 3.4587 3.0609 2.7403

15 6.8109 6.1422 5.8474 5.5755 5.0916 4.6755 4.0013 3.4834 3.0764 2.7502

16 6.9740 6.2651 5.9542 5.6685 5.1624 4.7296 4.0333 3.5026 3.0882 2.7575

17 7.1196 6.3729 6.0472 5.7487 5.2223 4.7746 4.0591 3.5177 3.0971 2.7629

18 7.2497 6.4674 6.1280 5.8178 5.2732 4.8122 4.0799 3.5294 3.1039 2.7668

19 7.3658 6.5504 6.1982 5.8775 5.3162 4.8435 4.0967 3.5386 3.1090 2.7697

20 7.4694 6.6231 6.2593 5.9288 5.3527 4.8696 4.1103 3.5458 3.1129 2.7718

21 7.5620 6.6870 6.3125 5.9731 5.3837 4.8913 4.1212 3.5514 3.1158 2.7734

22 7.6446 6.7429 6.3587 6.0113 5.4099 4.9094 4.1300 3.5558 3.1180 2.7746

23 7.7184 6.7921 6.3988 6.0442 5.4321 4.9245 4.1371 3.5592 3.1197 2.7754

24 7.7843 6.8351 6.4338 6.0726 5.4509 4.9371 4.1428 3.5619 3.1210 2.7760

25 7.8431 6.8729 6.4641 6.0971 5.4669 4.9476 4.1474 3.5640 3.1220 2.7765

26 7.8957 6.9061 6.4906 6.1182 5.4804 4.9563 4.1511 3.5656 3.1227 2.7768

27 7.9426 6.9352 6.5135 6.1364 5.4919 4.9636 4.1542 3.5669 3.1233 2.7771

28 7.9844 6.9607 6.5335 6.1520 5.5016 4.9697 4.1566 3.5679 3.1237 2.7773

29 8.0218 6.9830 6.5509 6.1656 5.5098 4.9747 4.1585 3.5687 3.1240 2.7774

30 8.0552 7.0027 6.5660 6.1772 5.5168 4.9789 4.1601 3.5693 3.1242 2.7775

40 8.2438 7.1050 6.6418 6.2335 5.5482 4.9966 4.1659 3.5712 3.1250 2.7778

50 8.3045 7.1327 6.6605 6.2463 5.5541 4.9995 4.1666 3.5714 3.1250 2.7778

60 8.3240 7.1401 6.6651 6.2492 5.5553 4.9999 4.1667 3.5714 3.1250 2.7778

ICFAI

Tables

103

STANDARD NORMAL PROBABILITY DISTRIBUTION TABLE

x

Z− µ

=σ

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753

0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621

1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830

1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015

1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177

1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817

2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857

2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890

2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916

2.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936

2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952

2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964

2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974

2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981

2.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4986

3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990

ICFAI

Formulae and Tables

104

t DISTRIBUTION TABLE

t = x − µ

σ

Degrees of freedom Area in both tails combined

0.10 0.05 .02 .01

1 6.314 12.706 31.821 63.657

2 2.920 4.303 6.965 9.925

3 2.353 3.182 4.541 5.841

4 2.132 2.776 3.747 4.604

5 2.015 2.571 3.365 4.032

6 1.943 2.447 3.143 3.707

7 1.895 2.365 2.998 3.499

8 1.860 2.306 2.896 3.355

9 1.833 2.262 2.821 3.250

10 1.812 2.228 2.764 3.169

11 1.796 2.201 2.718 3.106

12 1.782 2.179 2.681 3.055

13 1.771 2.160 2.650 3.012

14 1.761 2.145 2.624 2.977

15 1.753 2.131 2.602 2.947

16 1.746 2.120 2.583 2.921

17 1.740 2.110 2.567 2.898

18 1.734 2.101 2.552 2.878

19 1.729 2.093 2.539 2.861

20 1.725 2.086 2.528 2.845

21 1.721 2.080 2.518 2.831

22 1.717 2.074 2.508 2.819

23 1.714 2.069 2.500 2.807

24 1.711 2.064 2.492 2.797

25 1.708 2.060 2.485 2.787

26 1.706 2.056 2.479 2.779

27 1.703 2.052 2.473 2.771

28 1.701 2.048 2.467 2.763

29 1.699 2.045 2.462 2.756

30 1.697 2.042 2.457 2.750

40 1.684 2.021 2.423 2.704

60 1.671 2.000 2.390 2.660

120 1.658 1.980 2.358 2.617

Normal Distribution 1.645 1.960 2.326 2.576

ICFAI

Tables

105

AREA IN THE RIGHT TAIL OF A CHI-SQUARE (χχχχ2)

DISTRIBUTION TABLE

Degrees of freedom Area in right tail

0.99 0.975 0.95 0.90 0.80

1 0.00016 0.00098 0.00398 0.0158 0.0642

2 0.0201 0.0506 0.103 0.211 0.446

3 0.115 0.216 0.352 0.584 1.005

4 0.297 0.484 0.711 1.064 1.649

5 0.554 0.831 1.145 1.610 2.343

6 0.872 1.237 1.635 2.204 3.070

7 1.239 1.690 2.167 2.833 3.822

8 1.646 2.180 2.733 3.490 4.594

9 2.088 2.700 3.325 4.168 5.380

10 2.558 3.247 3.940 4.865 6.179

11 3.053 3.816 4.575 5.578 6.989

12 3.571 4.404 5.226 6.304 7.807

13 4.107 5.009 5.892 7.042 8.634

14 4.660 5.629 6.571 7.790 9.467

15 5.229 6.262 7.261 8.547 10.307

16 5.812 6.908 7.962 9.312 11.152

17 6.408 7.564 8.672 10.085 12.002

18 7.015 8.231 9.390 10.865 12.857

19 7.633 8.907 10.117 11.651 13.716

20 8.260 9.591 10.851 12.443 14.578

21 8.897 10.283 11.591 13.240 15.445

22 9.542 10.982 12.338 14.041 16.314

23 10.196 11.689 13.091 14.848 17.187

24 10.856 12.401 13.848 15.658 18.062

25 11.524 13.120 14.611 16.473 18.940

26 12.198 13.844 15.379 17.292 19.820

27 12.879 14.573 16.151 18.114 20.703

28 13.565 15.308 16.928 18.939 21.588

29 14.256 16.047 17.708 19.768 22.475

30 14.953 16.791 18.493 20.599 23.364

ICFAI

Formulae and Tables

106

AREA IN THE RIGHT TAIL OF A CHI-SQUARE (χχχχ2)

DISTRIBUTION TABLE

Note: If ν, the number of degrees of freedom, is greater than 30, we can approximate2αχ ,the

chi-square value leaving of the area in the right tail, by

3

2α α

2 2χ = ν 1 + z

9ν 9ν

−

where zα is the standard normal value that leaves α of the area in the right tail.

Area in right tail Degrees of freedom

0.20 0.10 0.05 0.025 0.01

1.642 2.706 3.841 5.024 6.635 1

3.219 4.605 5.991 7.378 9.210 2

4.642 6.251 7.815 9.348 11.345 3

5.989 7.779 9.488 11.143 13.277 4

7.289 9.236 11.070 12.833 15.086 5

8.558 10.645 12.592 14.449 16.812 6

9.803 12.017 14.067 16.013 18.475 7

11.030 13.362 15.507 17.535 20.090 8

12.242 14.684 16.919 19.023 21.666 9

13.442 15.987 18.307 20.483 23.209 10

14.631 17.275 19.675 21.920 24.725 11

15.812 18.549 21.026 23.337 26.217 12

16.985 19.812 22.362 24.736 27.688 13

18.151 21.064 23.685 26.119 29.141 14

19.311 22.307 24.996 27.488 30.578 15

20.465 23.542 26.296 28.845 32.000 16

21.615 24.769 27.587 30.191 33.409 17

22.760 25.989 28.869 31.526 34.805 18

23.900 27.204 30.144 32.852 36.191 19

25.038 28.412 31.410 34.170 37.566 20

26.171 29.615 32.671 35.479 38.932 21

27.301 30.813 33.924 36.781 40.289 22

28.429 32.007 35.172 38.076 41.638 23

29.553 33.196 36.415 39.364 42.980 24

30.675 34.382 37.652 40.647 44.314 25

31.795 35.563 38.885 41.923 45.642 26

32.912 36.741 40.113 43.194 46.963 27

34.027 37.916 41.337 44.461 48.278 28

35.139 39.087 42.557 45.722 49.588 29

36.250 40.256 43.773 46.979 50.892 30

ICFAI

Tables

107

F DISTRIBUTION TABLE

Degrees of Freedom for Numerator

Deg

rees

of Freed

om for D

enominator

1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞

1 161 200 216 225 230 234 237 239 241 242 244 246 248 249 250 251 252 253 254

2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.4 19.4 19.4 19.5 19.5 19.5 19.5 19.5 19.5

3 10.1 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.53

4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63

5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.37

6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67

7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23

8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93

9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71

10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.91 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54

11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.40

12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30

13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21

14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13

15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07

16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01

17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.96

18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.92

19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88

20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84

21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81

22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78

23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76

24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73

25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71

30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62

40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51

60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39

120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25

∞ 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00

ICFAI

Formulae and Tables

108

F DISTRIBUTION TABLE

Degrees of Freedom for Numerator

1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞

Deg

rees

of Freed

om for Den

ominator

1 4,052 5,000 5,403 5,625 5,764 5,859 5,928 5,982 6,023 6,056 6,106 6,157 6,209 6,235 6,261 6,287 6,313 6,339 6,366

2 98.5 99.0 99.2 99.2 99.3 99.3 99.4 99.4 99.4 99.4 99.4 99.4 99.4 99.5 99.5 99.5 99.5 99.5 99.5

3 34.1 30.8 29.5 28.7 28.2 27.9 27.7 27.5 27.3 27.2 27.1 26.9 26.7 26.6 26.5 26.4 26.3 26.2 26.1

4 21.2 18.0 16.7 16.0 15.5 15.2 15.0 14.8 14.7 14.5 14.5 14.4 14.2 14.0 13.9 13.8 13.7 13.7 13.6

5 16.3 13.3 12.1 11.4 11.0 10.7 10.5 10.3 10.2 10.1 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02

6 13.7 10.9 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88

7 12.2 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65

8 11.3 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86

9 10.6 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.31

10 10.0 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.91

11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.60

12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.36

13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.17

14 8.86 6.51 5.56 5.04 4.70 4.46 4.28 4.14 4.03 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.27 3.09 3.00

15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87

16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.75

17 8.40 6.11 5.19 4.67 4.34 4.10 3.93 3.79 3.68 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.65

18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.57

19 8.19 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.49

20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42

21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40 3.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.36

22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.31

23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 3.21 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.26

24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21

25 7.77 5.57 4.68 4.18 3.86 3.63 3.46 3.32 3.22 3.13 2.99 2.85 2.70 2.62 2.53 2.45 2.36 2.27 2.17

30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.01

40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.80

60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60

120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38

∞ 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00

ICFAI

Tables

109

CONTROL CHART FACTORS TABLE

Factors for x Charts Factors for R Charts

Sample Size, n σ

Rd2 =

nd

3A

2

2=

σ

σ

dR

3=

2

3

3 d

3d

1D −=

2

3

4 d

3d

1D +=

2 1.128 1.881 0.853 0 3.269

3 1.693 0.023 0.888 0 2.574

4 2.059 0.729 0.880 0 2.282

5 2.326 0.577 0.864 0 2.114

6 2.534 0.483 0.848 0 2.004

7 2,704 0.419 0.833 0.076 1.924

8 2.847 0.373 0.820 0.136 1.864

9 2.970 0.337 0.808 0.184 1.186

10 3.078 0.308 0.797 0.223 1.777

11 3.173 0.285 0.787 0.256 1.744

12 3.258 0.266 0.779 0.283 1.717

13 3.336 0.249 0.770 0.308 1.692

14 3.407 0.235 0.763 0.328 1.672

15 3.472 0.223 0.756 0.347 1.637

16 3.532 0.212 0.750 0.363 1.637

17 3.588 0.203 0.744 0.378 1.622

18 3.640 0.194 0.739 0.391 1.609

19 3,689 0.187 0.734 0.403 1.597

20 3.735 0.180 0.729 0.414 1.586

21 3,778 0.173 0.724 0.425 1.575

22 3.819 0.167 0.720 0.434 1.566

23 3.858 0.162 0.716 0.443 1.557

24 3.895 0.157 0.712 0.452 1.548

25 3.931 0.153 0.708 0.460 1.540

Note: If 1 – 3d3/d

2 < 0, then D

3 = 0.

ICFAI

Formulae and Tables

110

TABLE FOR VALUE OF CALL OPTION AS PERCENTAGE OF SHARE PRICE

Share Price Divided by PV (Exercise Price) Standard

Deviation Tim

es S

quare

Root of Tim

e

0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00

0.05 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.3 0.6 1.2 2.0

0.10 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.2 0.3 0.5 0.8 1.2 1.7 2.3 3.1 4.0

0.15 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.2 0.5 0.7 1.0 1.3 1.7 2.2 2.8 3.5 4.2 5.1 6.0

0.20 0.0 0.0 0.0 0.0 0.0 0.1 0.4 0.8 1.5 1.9 2.3 2.8 3.4 4.0 4.7 5.4 6.2 7.1 8.0

0.25 0.0 0.0 0.0 0.1 0.2 0.5 1.0 1.8 2.8 3.3 3.9 4.5 5.2 5.9 6.6 7.4 8.2 9.1 9.9

0.30 0.0 0.1 0.1 0.3 0.7 1.2 2.0 3.1 4.4 5.0 5.7 6.3 7.0 7.8 8.6 9.4 10.2 11.1 11.9

0.35 0.1 0.2 0.4 0.8 1.4 2.3 3.3 4.6 6.2 6.8 7.5 8.2 9.0 9.8 10.6 11.4 12.2 13.0 13.9

0.40 0.2 0.5 0.9 1.6 2.4 3.5 4.8 6.3 8.0 8.7 9.4 10.2 11.0 11.7 12.5 13.4 14.2 15.0 15.9

0.45 0.5 1.0 1.7 2.6 3.7 5.0 6.5 8.1 9.9 10.6 11.4 12.2 12.9 13.7 14.5 15.3 16.2 17.0 17.8

0.50 1.0 1.7 2.6 3.7 5.1 6.6 8.2 10.0 11.8 12.6 13.4 14.2 14.9 15.7 16.5 17.3 18.1 18.9 19.7

0.55 1.7 2.6 3.8 5.1 6.6 8.3 10.0 11.9 13.8 14.6 15.4 16.1 16.9 17.7 18.5 19.3 20.1 20.9 21.7

0.60 2.5 3.7 5.1 6.6 8.3 10.1 11.9 13.8 15.8 16.6 17.4 18.1 18.9 19.7 20.5 21.3 22.0 22.8 23.6

0.65 3.6 4.9 6.5 8.2 10.0 11.9 13.8 15.8 17.8 18.6 19.3 20.1 20.9 21.7 22.5 23.2 24.0 24.7 25.5

0.70 4.7 6.3 8.1 9.9 11.9 13.8 15.8 17.8 19.8 20.6 21.3 22.1 22.9 23.6 24.4 25.2 25.9 26.6 27.4

0.75 6.1 7.9 9.8 11.7 13.7 15.8 17.8 19.8 21.8 22.5 23.3 24.1 24.8 25.6 26.3 27.1 27.8 28.5 29.2

0.80 7.5 9.5 11.5 13.6 15.7 17.7 19.8 21.8 23.7 24.5 25.3 26.0 26.8 27.5 28.3 29.0 29.7 30.4 31.1

0.85 9.1 11.2 13.3 15.5 17.6 19.7 21.8 23.8 25.7 26.5 27.2 28.0 28.7 29.4 30.2 30.9 31.6 32.2 32.9

0.90 10.7 13.0 15.2 17.4 19.6 21.7 23.8 25.8 27.7 28.4 29.2 29.9 30.6 31.8 32.0 32.7 33.4 34.1 34.7

0.95 12.5 14.8 17.1 19.4 21.6 23.7 25.7 27.7 29.6 30.4 31.1 31.8 32.5 33.2 33.9 34.6 35.2 35.9 36.5

1.00 14.3 16.7 19.1 21.4 23.6 25.7 27.7 29.7 31.6 32.3 33.0 33.7 34.4 35.1 35.7 36.4 37.0 37.7 38.3

1.05 16.1 18.6 21.0 23.3 25.6 27.7 29.7 31.6 33.5 34.2 34.9 35.6 36.2 36.9 37.6 38.2 38.8 39.4 40.0

1.10 18.0 20.6 23.0 25.3 27.5 29.6 31.6 33.5 35.4 36.1 36.7 37.4 38.1 38.7 39.3 40.0 40.6 41.2 41.6

1.15 20.0 22.5 25.0 27.3 29.5 31.6 33.6 35.4 37.2 37.9 38.6 39.2 39.9 40.5 41.1 41.7 42.3 42.9 43.5

1.20 21.9 24.5 27.0 29.3 31.5 33.6 35.5 37.3 39.1 39.7 40.4 41.0 41.7 42.3 42.9 43.5 44.0 44.6 45.1

1.25 23.9 26.5 29.0 31.3 33.5 35.5 37.4 39.2 40.9 41.5 42.2 42.8 43.4 44.0 44.6 45.2 45.7 46.3 46.6

1.30 25.9 28.5 31.0 33.3 35.4 37.4 39.3 41.0 42.7 43.3 43.9 44.5 45.1 45.7 46.3 46.8 47.4 47.9 48.4

1.35 27.9 30.5 33.0 35.2 37.3 39.3 41.1 42.8 44.4 45.1 45.7 46.3 46.8 47.4 47.9 48.5 49.0 49.5 50.0

1.40 29.9 32.5 34.9 37.1 39.2 41.1 42.9 44.6 46.2 46.8 47.4 47.9 48.5 49.0 49.6 50.1 50.6 51.1 51.6

1.45 31.9 34.5 36.9 39.1 41.1 43.0 44.7 46.4 47.9 48.5 49.0 49.6 50.1 50.7 51.2 51.7 52.2 52.7 53.2

1.50 33.8 36.4 38.8 40.9 42.9 44.8 45.5 48.1 49.6 50.1 50.7 51.2 51.8 52.3 52.8 53.3 53.7 54.2 54.7

1.55 35.8 38.4 40.7 42.8 44.8 46.6 48.2 49.8 51.2 51.8 52.3 52.8 53.3 53.8 54.3 54.8 55.3 55.7 56.2

1.60 37.8 40.3 42.6 44.6 46.5 48.3 49.9 51.4 52.8 53.4 53.9 54.4 54.9 55.4 55.9 56.3 56.8 57.2 57.6

1.65 39.7 42.2 44.4 46.4 48.3 50.0 51.6 53.1 54.4 54.9 55.4 55.9 56.4 56.9 57.3 57.8 58.2 58.6 59.1

1.70 41.6 44.0 46.2 48.2 50.0 51.7 53.2 54.7 56.0 56.5 57.0 57.5 57.9 58.4 58.8 59.2 59.7 60.1 60.5

1.75 43.5 45.9 48.0 50.0 51.7 53.4 54.8 56.2 57.5 58.0 58.5 58.9 59.4 59.8 60.2 60.7 61.1 61.5 61.8

2.00 52.5 54.6 56.5 58.2 59.7 61.1 62.4 63.6 64.6 65.0 65.4 65.8 66.2 66.6 66.9 67.3 67.6 67.9 68.3

2.25 60.7 62.5 64.1 65.6 66.8 68.0 69.1 70.0 70.9 71.3 71.6 71.9 72.2 72.5 72.8 73.1 73.4 73.7 73.9

2.50 67.9 69.4 70.8 72.0 73.1 74.0 74.9 75.7 76.4 76.7 77.0 77.2 77.5 77.7 78.0 78.2 78.4 78.7 78.9

2.75 74.2 75.4 76.6 77.5 78.4 79.2 79.9 80.5 81.1 81.4 81.6 81.8 82.0 82.2 82.4 82.6 82.7 82.9 83.1

3.00 79.5 80.5 81.4 82.2 82.9 83.5 84.1 84.6 85.1 85.3 85.4 85.6 85.8 85.9 86.1 86.2 86.4 86.5 86.6

3.50 87.6 88.3 88.8 89.3 89.7 90.1 90.5 90.8 91.1 91.2 91.3 91.4 91.5 91.6 91.6 91.7 91.8 91.9 92.0

4.00 92.9 93.3 93.6 93.9 94.2 94.4 94.6 94.8 94.9 95.0 95.0 95.1 95.2 95.2 95.3 95.3 95.4 95.4 95.4

4.50 96.2 96.4 96.6 96.7 96.9 97.0 97.1 97.2 97.3 97.3 97.3 97.4 97.4 97.4 97.5 97.5 97.5 97.5 97.6

5.00 98.1 98.2 98.3 98.3 98.4 98.5 98.5 98.6 98.6 98.6 98.6 98.7 98.7 98.7 98.7 98.7 98.7 98.7 98.8

ICFAI

Tables

111

Share Price Divided by PV (Exercise Price)

Standard

Deviation Tim

es S

quare

Root of Tim

e

1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.75 2.00 2.50

0.05 3.1 4.5 6.0 7.5 9.1 10.7 12.3 13.8 15.3 16.7 20.0 23.1 25.9 28.6 31.0 33.3 42.9 50.0 60.0

0.10 5.0 6.1 7.3 8.6 10.0 11.3 12.7 14.1 15.4 16.8 20.0 23.1 25.9 28.6 31.0 33.3 42.9 50.0 60.0

0.15 7.0 8.0 9.1 10.2 11.4 12.6 13.8 15.0 16.2 17.4 20.4 23.3 26.0 28.6 31.1 33.3 42.9 50.0 60.0

0.20 8.9 9.9 10.9 11.9 13.0 14.1 15.2 16.3 17.4 18.5 21.2 23.9 26.4 28.9 31.2 33.5 42.9 50.0 60.0

0.25 10.9 11.8 12.8 13.7 14.7 15.7 16.7 17.7 18.7 19.8 22.3 24.7 27.1 29.4 31.7 33.8 42.9 50.0 60.0

0.30 12.8 13.7 14.6 15.6 16.5 17.4 18.4 19.3 20.3 21.2 23.5 25.8 28.1 30.2 32.3 34.3 43.1 50.1 60.0

0.35 14.8 15.6 16.5 17.4 18.3 19.2 20.1 21.0 21.9 22.7 24.9 27.1 29.2 31.2 33.2 35.1 43.5 50.2 60.0

0.40 16.7 17.5 18.4 19.2 20.1 20.9 21.8 22.6 23.5 24.3 26.4 28.4 30.4 32.3 34.2 36.0 44.0 50.5 60.1

0.45 18.6 19.4 20.3 21.1 21.9 22.7 23.5 24.3 25.1 25.9 27.9 29.8 31.7 33.5 35.3 37.0 44.6 50.8 60.2

0.50 20.5 21.3 22.1 22.9 23.7 24.5 25.3 26.1 26.8 27.6 29.5 31.3 33.1 34.8 36.4 38.1 45.3 51.3 60.4

0.55 22.4 23.2 24.0 24.8 25.5 26.3 27.0 27.8 28.5 29.2 31.0 32.8 34.5 36.1 37.7 39.2 46.1 51.9 60.7

0.60 24.3 25.1 25.8 26.6 27.3 28.1 28.8 29.5 30.2 30.9 32.6 34.3 35.9 37.5 39.0 40.4 47.0 52.5 61.0

0.65 26.2 27.0 27.7 28.4 29.1 29.8 30.5 31.2 31.9 32.6 34.2 35.8 37.4 38.9 40.3 41.7 48.0 53.3 61.4

0.70 28.1 28.8 29.5 30.2 30.9 31.6 32.3 32.9 33.6 34.2 35.8 37.3 38.8 40.3 41.6 43.0 49.0 54.0 61.9

0.75 29.9 30.6 31.3 32.0 32.7 33.3 34.0 34.6 35.3 35.9 37.4 38.9 40.3 41.7 43.0 44.3 50.0 54.9 62.4

0.80 31.8 32.4 33.1 33.8 34.4 35.1 35.7 36.3 36.9 37.5 39.0 40.4 41.8 43.1 44.4 45.6 51.1 55.8 63.0

0.85 33.6 34.2 34.9 35.5 36.2 36.8 37.4 38.0 38.6 39.2 40.6 41.9 43.3 44.5 45.8 46.9 52.2 56.7 63.6

0.90 35.4 36.0 36.6 37.3 37.9 38.5 39.1 39.6 40.2 40.8 42.1 43.5 44.7 46.0 47.1 48.3 53.3 57.6 64.3

0.95 37.2 37.8 38.4 39.0 39.6 40.1 40.7 41.3 41.8 42.4 43.7 45.0 46.2 47.4 48.5 49.6 54.5 58.6 65.0

1.00 38.9 39.5 40.1 40.7 41.2 41.8 42.4 42.9 43.4 44.0 45.2 46.5 47.6 48.8 49.9 50.9 55.6 59.5 65.7

1.05 40.6 41.2 41.8 42.4 42.9 43.5 44.0 44.5 45.0 45.5 46.8 48.0 49.1 50.2 51.2 52.2 56.7 60.5 66.5

1.10 42.3 42.9 43.5 44.0 44.5 45.1 45.6 46.1 46.6 47.1 48.3 49.4 50.5 51.6 52.6 53.5 57.9 61.5 67.2

1.15 44.0 44.6 45.1 45.6 46.2 46.7 47.2 47.7 48.2 48.6 49.8 50.9 51.9 52.9 53.9 54.9 59.0 62.5 68.0

1.20 45.7 46.2 46.7 47.3 47.8 48.3 48.7 49.2 49.7 50.1 51.3 52.3 53.3 54.3 55.2 56.1 60.2 63.5 68.8

1.25 47.3 47.8 48.4 48.8 49.3 49.8 50.3 50.7 51.2 51.6 52.7 53.7 54.7 55.7 56.6 57.4 61.3 64.5 69.6

1.30 48.9 49.4 49.9 50.4 50.9 51.3 51.8 52.2 52.7 53.1 54.1 55.1 56.1 57.0 57.9 58.7 62.4 65.5 70.4

1.35 50.5 51.0 51.5 52.0 52.4 52.9 53.3 53.7 54.1 54.6 55.6 56.5 57.4 58.3 59.1 59.9 63.5 66.5 71.1

1.40 52.1 52.6 53.0 53.5 53.9 54.3 54.8 55.2 55.6 56.0 56.9 57.9 58.7 59.6 60.4 61.2 64.6 67.5 71.9

1.45 53.6 54.1 54.5 55.0 55.4 55.8 56.2 56.6 57.0 57.4 58.3 59.2 60.0 60.9 61.6 62.4 65.7 68.4 72.7

1.50 55.1 55.6 56.0 56.4 56.8 57.2 57.6 58.0 58.4 58.8 59.7 60.5 61.3 62.1 62.9 63.6 66.8 69.4 73.5

1.55 56.6 57.0 57.4 57.8 58.2 58.6 59.0 59.4 59.7 60.1 61.0 61.8 62.6 63.3 64.1 64.7 67.8 70.3 74.3

1.60 58.0 58.5 58.9 59.2 59.6 60.0 60.4 60.7 61.1 61.4 62.3 63.1 63.8 64.5 65.2 65.9 68.8 71.3 75.1

1.65 59.5 59.9 60.2 60.6 61.0 61.4 61.7 62.1 62.4 62.7 63.5 64.3 65.0 65.7 66.4 67.0 69.9 72.2 75.9

1.70 60.9 61.2 61.6 62.0 62.3 62.7 63.0 63.4 63.7 64.0 64.8 65.5 66.2 66.9 67.5 68.2 70.9 73.1 76.6

1.75 62.2 62.6 62.9 63.3 63.6 64.0 64.3 64.6 64.9 65.3 66.0 66.7 67.4 68.0 68.7 69.2 71.9 74.0 77.4

2.00 68.6 68.9 69.2 69.5 69.8 70.0 70.3 70.6 70.8 71.1 71.7 72.3 72.9 73.4 73.9 74.4 76.5 78.3 81.0

2.25 74.2 74.4 74.7 74.9 75.2 75.4 75.6 75.8 76.0 76.3 76.8 77.2 77.7 78.1 78.5 78.9 80.6 82.1 84.3

2.50 79.1 79.3 79.5 79.7 79.9 80.0 80.2 80.4 80.6 80.7 81.1 81.5 81.9 82.2 82.6 82.9 84.3 85.4 87.2

2.75 83.3 83.4 83.6 83.7 83.9 84.0 84.2 84.3 84.4 84.6 84.9 85.2 85.5 85.8 86.0 86.3 87.4 88.3 89.7

3.00 86.8 86.9 87.0 87.1 87.3 87.4 87.5 87.6 87.7 87.8 88.1 88.3 88.5 88.8 89.0 89.2 90.0 90.7 91.8

3.50 92.1 92.1 92.2 92.3 92.4 92.4 92.5 92.6 92.6 92.7 92.8 93.0 93.1 93.3 93.4 93.5 94.0 94.4 95.1

4.00 95.5 95.5 95.6 95.6 95.7 95.7 95.7 95.8 95.8 95.8 95.9 96.0 96.1 96.2 96.2 96.3 96.6 96.8 97.2

4.50 97.6 97.6 97.6 97.6 97.7 97.7 97.7 97.7 97.8 97.8 97.8 97.9 97.9 97.9 98.0 98.0 98.2 98.3 98.5

5.00 98.8 98.8 98.8 98.8 98.8 98.8 98.8 98.8 98.9 98.9 98.9 98.9 98.9 99.0 99.0 99.0 99.1 99.1 99.2

ICFAI

Formulae and Tables

112

Table for N(x) When x ≤≤≤≤ 0

This table shows values of N(x) for x ≤ 0. The table should be used with interpolation. For example,

N(–0.1234) = N(–0.12) – 0.34[N(–0.12) – N(–0.13)]

= 0.4522 – 0.34 × (0.4522 – 0.4483)

= 0.4509 x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

–0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641

–0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247

–0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859

–0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483

–0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121

–0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776

–0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451

–0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148

–0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867

–0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611

–1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379

–1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170

–1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985

–1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823

–1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681

–1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559

–1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455

–1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367

–1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294

–1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233

–2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183

–2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143

–2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110

–2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084

–2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064

–2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048

–2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036

–2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026

–2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019

–2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014

–3.0 0.0014 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010

–3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007

–3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005

–3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003

–3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002

–3.5 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002

–3.6 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

–3.7 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

–3.8 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

–3.9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

–4.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

ICFAI

Tables

113

Table for N(x) When x ≥≥≥≥ 0

This table shows values of N(x) for x ≥ 0. The table should be used with interpolation. For example,

N(0.6278) = N(0.62) + 0.78[N(0.63) – N(0.62)]

= 0.7324 + 0.78 × (0.7357 – 0.7324)

= 0.7350 x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753

0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133

0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177

1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545

1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633

1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706

1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857

2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890

2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916

2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952

2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964

2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974

2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981

2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

3.0 0.9986 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993

3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995

3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997

3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998

3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

4.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

ICFAI

Formulae and Tables

114

TABLE FOR RELATIONSHIP BETWEEN NOMINAL AND

EFFECTIVE RATES OF INTEREST AND DISCOUNT

i(p)

= [(1 + i)1/p

– 1]p d(p)

= [1 – (1 – d)1/p

]p

d = i

1 i+

Effective rate

Interest

i(2) i(4) i(12) d d(2) d(4) d(12)

0.01 0.0100 0.0100 0.0100 0.0099 0.0099 0.0099 0.0099

0.02 0.0199 0.0199 0.0198 0.0196 0.0197 0.0198 0.0198

0.03 0.0298 0.0297 0.0296 0.0291 0.0293 0.0294 0.0295

0.04 0.0396 0.0394 0.0393 0.0385 0.0388 0.0390 0.0392

0.05 0.0494 0.0491 0.0489 0.0476 0.0482 0.0485 0.0487

0.06 0.0591 0.0587 0.0584 0.0566 0.0574 0.0578 0.0581

0.07 0.0688 0.0682 0.0678 0.0654 0.0665 0.0671 0.0675

0.08 0.0785 0.0777 0.0772 0.0741 0.0755 0.0762 0.0767

0.09 0.0881 0.0871 0.0865 0.0826 0.0843 0.0853 0.0859

0.10 0.0976 0.0965 0.0957 0.0909 0.0931 0.0942 0.0949

0.11 0.1071 0.1057 0.1048 0.0991 0.1017 0.1030 0.1039

0.12 0.1166 0.1149 0.1139 0.1071 0.1102 0.1117 0.1128

0.13 0.1260 0.1241 0.1228 0.1150 0.1186 0.1204 0.1216

0.14 0.1354 0.1332 0.1317 0.1228 0.1268 0.1289 0.1303

0.15 0.1448 0.1422 0.1406 0.1304 0.1350 0.1373 0.1390

0.16 0.1541 0.1512 0.1493 0.1379 0.1430 0.1457 0.1475

0.17 0.1633 0.1601 0.1580 0.1453 0.1510 0.1540 0.1560

0.18 0.1726 0.1690 0.1667 0.1525 0.1589 0.1621 0.1644

0.19 0.1817 0.1778 0.1752 0.1597 0.1666 0.1702 0.1727

0.20 0.1909 0.1865 0.1837 0.1667 0.1743 0.1782 0.1809

0.21 0.2000 0.1952 0.1921 0.1736 0.1818 0.1861 0.1891

0.22 0.2091 0.2039 0.2005 0.1803 0.1893 0.1940 0.1972

0.23 0.2181 0.2125 0.2088 0.1870 0.1967 0.2017 0.2052

0.24 0.2271 0.2210 0.2171 0.1935 0.2039 0.2094 0.2132

0.26 0.2450 0.2379 0.2334 0.2063 0.2183 0.2246 0.2289

0.28 0.2627 0.2546 0.2494 0.2188 0.2322 0.2394 0.2443

0.30 0.2804 0.2712 0.2653 0.2308 0.2459 0.2539 0.2595

0.32 0.2978 0.2875 0.2809 0.2424 0.2592 0.2682 0.2744

0.34 0.3152 0.3036 0.2963 0.2537 0.2723 0.2822 0.2891

0.36 0.3324 0.3196 0.3115 0.2647 0.2850 0.2960 0.3036

0.38 0.3495 0.3354 0.3264 0.2754 0.2975 0.3095 0.3178

0.40 0.3664 0.3510 0.3412 0.2857 0.3097 0.3227 0.3318

ICFAI

Tables

115

TABLE FOR RELATIONSHIP BETWEEN NOMINAL AND

EFFECTIVE RATES OF INTEREST AND DISCOUNT

I/effective rate

Interest

i/i(2) i/i(4) i/i(12) i/d(2) i/d(4) i/d(12)

0.01 1.0025 1.0037 1.0046 1.0075 1.0062 1.0054

0.02 1.0050 1.0075 1.0091 1.0150 1.0125 1.0108

0.03 1.0074 1.0112 1.0137 1.0224 1.0187 1.0162

0.04 1.0099 1.0149 1.0182 1.0299 1.0249 1.0215

0.05 1.0123 1.0186 1.0227 1.0373 1.0311 1.0269

0.06 1.0148 1.0222 1.0272 1.0448 1.0322 1.0372

0.07 1.0172 1.0259 1.0317 1.0522 1.0434 1.0375

0.08 1.0196 1.0295 1.0362 1.0596 1.0495 1.0428

0.09 1.0220 1.0331 1.0406 1.0670 1.0556 1.0481

0.10 1.0244 1.0368 1.0450 1.0744 1.0618 1.0534

0.11 1.0268 1.0404 1.0495 1.0818 1.0679 1.0586

0.12 1.0292 1.0439 1.0539 1.0892 1.0739 1.0639

0.13 1.0315 1.0475 1.0583 1.0965 1.0800 1.0691

0.14 1.0339 1.0511 1.0626 1.1039 1.0861 1.0743

0.15 1.0362 1.0546 1.0670 1.1112 1.0921 1.0795

0.16 1.0385 1.0581 1.0714 1.1185 1.0981 1.0847

0.17 1.0408 1.0617 1.0757 1.1258 1.1042 1.0899

0.18 1.0431 1.0652 1.0800 1.1331 1.1102 1.0950

0.19 1.0454 1.0687 1.0843 1.1404 1.1162 1.1002

0.20 1.0477 1.0722 1.0887 1.1477 1.1222 1.1053

0.21 1.0500 1.0756 1.0929 1.1550 1.1281 1.1104

0.22 1.0523 1.0791 1.0972 1.1623 1.1341 1.1155

0.23 1.0545 1.0825 1.1015 1.1695 1.1400 1.1206

0.24 1.0568 1.0860 1.1057 1.1768 1.1460 1.1257

0.26 1.0612 1.0928 1.1142 1.1912 1.1578 1.1359

0.28 1.0657 1.0996 1.1226 1.2057 1.1696 1.1460

0.30 1.0701 1.1064 1.1310 1.2201 1.1814 1.1560

0.32 1.0745 1.1131 1.1393 1.2345 1.1931 1.1660

0.34 1.0788 1.1197 1.1476 1.2488 1.2047 1.1759

0.36 1.0831 1.1264 1.1559 1.2631 1.2164 1.1859

0.38 1.0874 1.1330 1.1641 1.2774 1.2280 1.1957

0.40 1.0916 1.1395 1.1722 1.2916 1.2395 1.2055

ICFAI

116

Formulae Index

Accounting rate of return 41

Accumulated Value of

- deferred annuity certain 2

- deferred annuity due 3

- annuity due 3, 4, 6, 7, 12, 15-16, 19

- annuity 2, 3, 4-7, 10,12-16, 19

- immediate annuity 2, 3, 5-7, 12, 14,

16, 19

- increasing annuity 4, 5

- increasing annuity due 4

- increasing immediate annuity 5,6

Add on yield 55

Alpha 30

Altman’s Z score model 91

Amount of level premium to be paid 19

Annualized percentage premium 50

Annuity

- accumulated value 1-7

- present value 2-7, 11-16, 19

- net single premium 14

ANOVA 83

APV of a foreign project 51

Arbitrage possibility 50

Arithmetic mean 73, 74, 76

Asset beta 67

Average fixed cost

Average interest yield on the life fund 8

Average of relatives method 82

Average product of labor 22

Average propensity 25

- to consume 25

- to save 25

Basis point value 45

Baumol cash management model 41

Benefit-cost ratio 42

Beta of

- asset 67

- security 29

BHW model 54

Binomial distribution 77

Binomial option pricing model 46, 67

Black-Scholes option pricing model 46,68

Bond

- valuation 31

- change in value of 45

- duration of a perpetual 86

- duration on selling at par 85

- present value of 85

- transaction price of 44, 67

- treasury implied repo rate 45

Bower’s model 54

Break-even

- point 58

- output 23

- financial 35

- operating 35

- period 88

- size of investment 52

Budget constraint 21

B U&& HLAMANN Credibility 17

Call option

- delta of 48

- gamma of 49

- pay-off from buying an option 45

- parity equation 46

- rho for a european option 49

- theta of 48-49

- vega of 49

Capital account balance 26

Capital recovery factor 2, 28

Capital redemption policies 7

Cash flow from the view point of

- long-term funds 66

- equity 66

Cash management

- Baumol model 41, 90

- Miller and Orr model 41, 91

Cash price of futures 44

Central death rate 10

Chain index numbers 82

Change in value of a bond 45

Chi-square statistic for a sample variance 84

Chi-square statistic 83, 84

Children’s Deferred assurances 14

ICFAI

117

Coefficient

- of determination 79

- of multiple correlation 80

- of variation 75

- Karl Pearson’s correlation 79

- Rank correlation 79

Combined standard deviation of two

groups 75

Commutation Functions 11

Conn & Nielson model 92

Consumer credit 56

Consumer equilibrium 21

Conversion parity price 88

Conversion premium 88

Corporate dividend behavior 38

Correlation Co-efficient 63

Cost constraint of a firm 22

Cost of

- debentures 35

- equity capital 36-38

- preference capital 35-36

- term loans 35

Cost performance index 68-69

Covariance 29, 76, 61, 63, 90

Coverage ratios 32

Cross price elasticity of demand 20

Crude death rate 8

Current account balance 26

Current yield 31, 85, 86, 88

Cyclical variation 81

Daily volatility 49

Deferred

- accumulated value 1,2

- present value 2-7, 10-16

- children’s assurances 14

Definite integral 72

Degree of

- financial leverage 34

- operating leverage 34

- total leverage 34

Degrees of freedom in a contingency

table 83

Delta for

- call 48

- put 48

- portfolio of derivatives 48

Dependency ratio 9

Discriminate analysis 90

Dividend Policy

- corporate dividend behavior 38

- Gordon model 37

- MM approach 38

- traditional model 37

- Walter model 37

Domestic resource cost 67

Doubling period 1

Duration

- at various stages of production 38

- of perpetual bond 86

- of bond selling at par 86

- of equity 87

- limiting value of 86

- modified 87

- number of future contract 45, 65

- simplified formula 86

EBIT – EPS indifference point 89

Economic order quantity 39

Effect of changing the credit variables 40

Effective price in futures 43

Effective rate of interest 28, 56

Effective rate of protection 67

Effective vs. nominal rate of interest 1

Efficient input combination 22

Elasticity

- cross 20

- income 20

- interest rate 86

- price 20

- supply 23

- promotional 21

Empirical mode 74

Equilibrium

- consumer 21

- income 25

- goods market 25

- money market 26

Equity valuation 31

Equivalent loan model 53

Error sum of squares 79

Estimated

- cost performance index 68, 69

- probability of deaths 9

- return on a stock 43

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118

Expectations of life 10

Expected

- return and variance 43

- aggregate loss 18

- net present value 67

- standard deviation 67

- rate of return 31

- return of a portfolio 62

- arbitrage pricing theory 61

- time in project scheduling 68

- utility of asset mix 62

- value 75

Exponential distribution 9

Extended probabilistic analysis 89

External financing requirement 33

Factors influencing option prices 46

Fertility rates 8

Finance interrelation ratio 26

Finance ratio 26

Financial break-even point 35

Fiscal deficit 27

Fisher’s ideal

- price index 81

- quantity index 81

Future Value 1, 28

Futures

- effective price 43

- margin 43

- number contracts 65

- cash price 43

Gamma distribution 18

Gamma of call or put 49

Geometric mean 74

GNP deflator 24

Goods market equilibrium 25

Gross yield 55

Harmonic mean 74

Hedge Ratio 44

Herfindahl’s index 24

High powered money 26

Hire Purchase

- finance company’s angle 56

- hirer’s angle 55

Historical (ex post)

- return 43

- variance 43

- standard deviation 43

Holding period yield 62

Housing finance

- disbursement amount 57

- equated monthly installments 57

Hypergeometric distribution 77

Implied repo rate 44, 45

Income elasticity of demand 20

Index

- chain 82

- cost performance 68, 69

- estimated cost performance 69

- Fisher’s price and quantity 82

- Laspeyre’s price and quantity 81

- Lerner monopoly power 24

- Marshall Edgeworth price 82

- odd-lot 87, 88

- Paasche’s price 81

- schedule performance 68

- Herfindahl’s 24

- Laspeyre price 24

- unweighted aggregates price 81

- value 81

Inference about two population variances 84

Integration 71

Interest rate

- elasticity 86

- parity 50

- risk 86

Interest rebate 56

Intermediation ratio 26

Internal rate of return 37, 42, 55, 66

- after tax cost of leasing 55

- implied repo rate 44

- based pricing 55

- Modified 66, 86

Interpolation and Extrapolation 72

Intrinsic value or present value of

- equity share 87

- bond 85

Jensen’s differential return 64

Karl Pearson’s correlation coefficient 79

Labor

- cost variance 59, 60

- efficiency variance 59, 60

- efficiency sub-variance 60

- mix variance 59, 60

- rate variance 59, 60

- yield variance 59, 60

ICFAI

119

Laspeyre’s

- price index 24, 81, 82

- quantity index 82

Lerner index of monopoly power 24

Leverage

- ratios 32

- financial 34

- operating 34

- total 34

Limiting value of duration 86

Liquidity ratios 32

Loading profit that is profit due to lower

expenses 19

Logarithms 70

Lognormal distribution 17

Margin 59

- futures 59

- of safety 59

Marginal

- cost 23, 24

- product of labor 22

- propensity to consume 25

- rate of substitution 21

- rate of technical substitution 22

- revenue 23, 24

Market price of the property 88

Marriage rates 8

Marshall Edgeworth price index 82

Material

- cost variance 59

- mix variance 59

- price variance 59

- usage variance 59

- sub-usage variance 59

- yield variance 59

Mean

- arithmetic 73, 74

- geometric 74

- harmonic 74

- collective risk model 18

- absolute deviation 74

- weighted arithmetic 73

- weighted harmonic 74

Median 73

Migration rates of area 9

Miller and Orr model 41, 91

Minimum variance hedge ratio 44

Mode 74

Models

- Altman’s Z score 91

- Baumol cash management 41, 90

- Binomial option pricing 41, 90

- Black-Scholes option pricing 46, 98

- BHW 54

- Bower’s 54

- cash management 41, 90

- Conn & Nielson 92

- Gordon dividend policy model 37

- MM dividend policy 37

- traditional dividend policy 37

- Walter dividend policy 37

- Equivalent loan 53

- individual risk 18

- collective risk 18

- Miller and Orr cash management 41

- CAPM 29, 43, 52, 61

- Weingartner’s 53

Modified

- duration 86

- net present value 66

- internal rate of return 66

Money market equilibrium 26

Money supply 26

Multiple regression equation 80

Multiplier

- income 26

- money 26

Net acquisition value 91

Net advantage of leasing (NAL) 54

Net annual premium 14

Net Asset Value 88

Net operating cycle period 38

Net present value 41, 42, 57, 66

- modified 66

- venture capital 57

- expected 67

Net-benefit-cost ratio 42

New issue ratio 26

Nominal rate 28

Number of futures contracts 65

Odd-lot

- index 87

- short sales ratio 88

ICFAI

120

Office premium 8, 16, 19

Operating break-even point 35

Operating cycle

- Net 38

- Weighted 38

Optimal portfolio selection using sharpe’s

optimization 62

Option pricing

- Binomial model 46, 67

- Black-Scholes model 49

Output determination in oligopoly 24

Overall break even point 35

Overall capitalization rate of the firm 36

p Charts 83

Paasche’s price index 81

Pareto distribution 18

Partial derivatives 71

Pay-off from Buying a

- call option 45

- put option 45

Payback period 88

Percentage of downside risk 88

Percentage price volatility 86

Permutations and combinations 70

Poisson distribution 16, 17, 77

Policy value for a whole life assurance

policy 19

Policy value under prospective method 19

Population standard deviation 74

Portfolio insurance 49

Premiums

- office 8, 16, 19

- for additional risks 15

- when frequency of payment is

m times a year 15

Present Value of

- deferred annuity 2, 3, 14

- deferred annuity certain 2

- deferred annuity due 3

- deferred life annuity 12

- deferred perpetuity 3

- deferred temporary immediate

life annuity 13

- life annuity 13

- perpetuity 2

- perpetuity due 3, 4

- temporary immediate life annuity 12, 13

- annuity due 2-4, 6, 7, 12, 15, 16, 19

- immediate annuity certain 2, 3

- immediate annuity 2, 3, 5-7, 12, 14,

16, 19

- immediate increasing perpetuity 4

- immediate increasing annuity 4

- immediate perpetuity 3

- increasing annuity 4, 5

- increasing annuity due 4

- increasing life annuity in terms of

commutation function 13

- increasing perpetuity due 4

- increasing whole life assurance 11, 12

- assurance benefits to the insured

in terms of the commutation

functions 11

- benefits 10, 11, 19

- rental stream 54

- perpetuity 2-4, 28

- annuity 15

- decreasing term assurance policy 14

- term assurance 10

- standard for full credibility for

severity 17

- interest tax shield 53, 54

Price elasticity of

- demand 20

- supply 21

Primary deficit 27

Probability 75

Probability that a person of age x years

- dies in another n years 10

- dies within one year 9, 10

- dies within the next n years 10

- survives another one year 10

- survives another n years 10

- will die within n years following

m years from now 10

Process variance for pure premium 17

Product of labor

- average product of labor 22

- marginal product of labor 22

Profit maximization in

- monopoly 23

- perfect competition 23

Profit of a firm 23

Profit/Volume ratio 58

Profitability ratios 33

Progressions 70

Promotional elasticity of demand 21

ICFAI

121

Propensity

- to consume, save (average) 25

- to consume, save (marginal) 25

Put

- delta for 48

- gamma of 49

- pay-off from buying an option 45

- parity equation 46

- rho for a European option 49

- theta of 48, 49

- vega of 49

Quartile deviation 74

R Charts 83, 109

Rank correlation coefficient 79

Rate of return 29

Ratios

- benefit-cost 42

- coverage 32

- dependency 9

- finance interrelation 26

- finance 26

- hedge 44, 45, 67

- intermediation 26

- leverage 32

- liquidity 32

- minimum variance hedge 44

- net-benefit-cost 42

- new issue 26

- odd-lot short sales 88

- profit/volume 58

- profitability 33

- Sharpe’s 62, 64

- Treynor’s 64

- turnover 33

- benefit-cost 42

Realized yield 85

Regression line 79

Regression sum of squares 79

Relation between

- EBIT and EPS 89

- ROI and ROE 89

Reorder point 39

Return

- accounting rate of 41

- estimated 43

- expected (of security) 30

- expected (of portfolio) 61

- under arbitrage pricing theory 61

- Historical (ex post) 43

- Internal rate or cost of leasing 54, 55

- Internal rate of return 37, 42

- Jensen’s differential 64

- under CAPM model 29, 43, 52, 61

- from net selectivity 64

- from total selectivity 64

Revenue deficit 27

Rho for a European

- call option 49

- put option 49

RS and RSI 121

Rules of differentiation 70

Sample standard deviation 75, 78, 79

Schedule performance index 68

Secular trend 80

Security

- Beta of 29

- Systematic risk of 30

- Unsystematic risk of 63

Share price Ex-Rights 53

Sharpe’s ratio 64

Simplified formula for duration 85

Sinking fund factor 1, 28

Standard deviation 17, 29, 43, 44, 49, 63,

67, 68, 75, 79, 82, 110, 111

- combined (two groups) 75, 78, 104

- expected 75

- historical 43

- population 74

- sample 74

Standard error 78-80

- a simple regression equation 79

- a multiple regression equation 80

Standard normal variable 77

Stochastics 88

Suggested framework for lease evaluation 54

Sustainable growth rate 34

Systematic risk of

- security 30

- portfolio 63

Tax-adjusted CAPM 61

Tax burden on the buyer 23

T-bill purchase price 44

t-distribution 78

Theta of

- call 48

- put 49

ICFAI

122

Total

- cost 22

- portfolio variance 63

- sum of squares 79

Trade balance 26

Transaction price of the bond 44, 67

Treasury bond implied repo rate 45

Treynor’s ratio 64

Turnover of primary/satellite dealer 53

Turnover ratios 33

Unsystematic risk of a security and

portfolio 30, 62, 63

Unweighted aggregates price index 81

Valuation

- bond 31

- equity 31

- convertible 32

- currency swaps 48

- firms 91

- interest rate swaps 48

Value index number 82

Value of

- right 53

- share after the rights issue 53

Variance

- chi-square statistic for a sample

variance 84

- covariance 90

- Expected 43, 61, 63

- historical (ex-post) 43

- inference about two population 84

- labor cost 59, 60

- labor efficiency 60

- labor efficiency sub 60

- labor mix 60

- labor rate 60

- labor yield 60

- material cost 59

- material mix 59

- material price 59

- material usage 59

- material sub-usage 59

- material yield 59

- minimum hedge ratio 44

- process (for pure premium) 17

- portfolio 65, 67

- in collective risk model 18

- in individual risk model 18

- in project scheduling 68

- portfolio 67

- asset 29

- returns on foreign investment 51

Vega of call or put 49

Velocity of money 26

Weibull distribution 9

Weighted

- arithmetic mean 73

- average cost of capital 36

- harmonic mean 74

- operating cycle 38

Weingartner’s model 53

X - Charts 82

Yield

- current 88

- gross 55

- realized 85

- average interest (on the life fund) 8

- holding period 29, 62

- labor variance 60

- material variance 59

- to maturity 31, 32, 45, 85, 86

Formulae And Tables - MBAflp.iutripura.edu.in/FAT Book.pdf · Phone : (+91) (040) 23430 – 368,...

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